MC-NRLF 


MENTAL 

JPfcl     I     O  iff  £Lr     JL    *' %** 


«# 


|U 


#? 


•    BOOK  «  COM  B\  NY 
YORK-  QNCINNATI-CHCCAGO     I 


IJMtfljfiftit! 


COMPLIMENTS 

AMERICAN  BOOK  CO, 

A.  P.  aUNN,  Gen'l  Ag't 
PIN  E&  BATTERY 

SAN   FRANCISCO. 


^ 


MENTAL    ARITHMETIC 


BY 

i.  c.  mcneill 

PRESIDENT   SEVENTH  WISCONSIN   STATE   NORMAL   SCHOOL 


>*^c 


NEW  YORK-:.  CINCINNATI-:.  CHICAGO 

AMERICAN    BOOK    COMPANY 


QlMc 

M  \  ^ 


COPTEIGHT,   1902,   BY 

i.  c.  McNeill. 

Entered  at  Stationers'  Hall,  London. 


MENTAL   ARITH. 


EDUCATION  £>&% 


PREFACE 

This  work  is  designed  for  use  in  grammar  grades.  It 
may  also  be  used  with  advantage  for  review  work  in  high 
schools  and  normal  schools.  Mental  or  oral  arithmetic 
when  properly  developed  and  taught  not  only  does  more 
than  any  other  agency  in  giving  pupils  insight  into  the  real 
nature  of  number  and  numerical  relations,  but  it  also  adds 
keenness  to  the  mind  in  general  and  strengthens  its  power 
of  concentration.  To  be  thus  efficacious,  however,  any 
course  in  mental  arithmetic  must  be  based  upon  sound  peda- 
gogic principles  and  proceed  along  sound  pedagogic  lines. 

The  principles  upon  which  this  book  is  based  may  be 
briefly  formulated.  Observation  must  precede  compari- 
son ;  sense  impressions  must  come  before  thought  rela- 
tions can  be  established ;  rational  movement  proceeds 
from  the  known  to  the  related  unknown,  from  the  simple 
to  the  complex,  and  from  the  particular  notion  to  the 
concept.  The  method  of  this  book  recognizes  the  impor- 
tant truth  that  first  perceptions  must  be  clear,  distinct, 
and  vivid  in  order  to  leave  in  the  mind  correct  traces  of 
ideas  through  which  the  later  impressions  may  be  apper- 
ceived.  The  value  of  clear  images  in  bringing  before  the 
mind  the  material  for  reasoning  and  in  leading  gradually 
to  the  formation  of  accurate  concepts  is  duly  emphasized. 

The  problems  have  been  specially  prepared  to  illustrate 
and  call  forth  ideas.  The  first  problem  looks  to  the  last 
through  all   that  come   between.      The  development  of 

541529 


•  4  •  :  ,'<-"':  '  . «"  '    '../-''*"• 4     PREFACE 

each  section  furnishes  a  foundation  upon  which  the  new 
thoughts  that  immediately  follow  may  most  readily  erect 
themselves.  A  constantly  increasing  demand  upon  the 
pupil's  mental  activity  goes  hand  in  hand  with  the  pro- 
gressive gain  in  power.  But  there  has  been  rigid  exclu- 
sion of  problems  of  so  unnatural  a  complexity  as  to 
impede  instead  of  furthering  the  free  movement  of  the 
mind. 

The  author  acknowledges  his  indebtedness  to  Professor 
C.  W.  Smith  of  the  Superior  State  Normal  School  for 
much  valuable  aid  in  the  preparation  of  the  problems  in 
this  book,  and  for  testing  in  preparatory  classes  every 
step  taken  in  the  development  of  the  subjects. 


SUGGESTIONS  TO  TEACHERS 

(a)  How  to  prepare  the  lesson  is  the  most  important  question  for 
the  pupils.  They  cannot  work  intelligently  unless  they  understand 
the  aim  of  each  day's  study.  The  great  function  of  the  teacher  is  to. 
guide  activity.  Learning  is  the  pupils'  act.  Before  an  intelligent 
assignment  of  the  lesson  can  be  made  by  the  teacher,  he  must  con- 
sider carefully  just  what  steps  are  new  and  how  far  the  pupils  are 
prepared  by  what  they  already  know  for  the  advanced  work.  In 
assigning  the  lesson,  the  teacher  should  impress  upon  his  pupils  the 
particular  end  in  view,  and  should  make  sure  that  they  understand 
just  what  they  are  to  accomplish  and  by  what  plans  and  devices 
they  may  best  succeed.  The  next  day's  recitation  will  test  the 
pupils'  understanding  of  such  directions  and  their  faithfulness  in 
following  them.  In  determining  the  pupils'  preparation  to  begin  a 
new  line  of  work,  the  skilled  teacher  will  approach  the  learners  on 
their  highest  plane  of  old  work,  directly  related  to  the  new  material 
to  be  considered.  If  students  move  from  this  position  with  ease  and 
freedom,  it  is  good  evidence  that  the  lower  phases  of  the  subject  are 
well  organized  in  their  minds.  If  they  do  not  show  a  mastery  here, 
it  is  well  to  descend  to  the  next  lower  phase  or  to  a  place  where  they 
are  able  to  stand  firm.  The  higher  phases  of  the  old  work  should 
be  brought  clearly  within  the  field  of  conscious  comprehension  before 
attempting  to  present  new  ideas. 

(6)  As  a  rule  it  is  a  good  plan  to  leave  the  diagrams  on  the  board 
for  constant  use  until  pupils  can  image  them.  When  the  diagrams 
can  be  held  in  the  mind,  it  is  not  productive  of  attention  or  study  to 
have  them  before  the  pupils.  In  their  study  it  is  of  great  value  to 
have  pupils  picture  or  represent  in  diagrams,  if  possible,  the  condi- 
tions of  new  problems.  Mensuration  is  not  comprehended  until 
pupils  through  practice  and  tests  perceive  the  relations. 

(c)  Good  results  may  be  reached  in  class  by  asking  the  pupils  with 

5 


6  SUGGESTIONS   TO   TEACHERS 

books  closed  to  work  the  problem  given  and  rise  at  a  signal  from  the 
teacher  when  they  have  finished.  This  reveals  at  a  glance  the  back- 
ward students  who  need  the  most  assistance.  Analyses  should  be  given 
in  simple,  direct  sentences.  It  is  often  a  good  plan  to  have  one  pupil 
take  up  the  analysis  when  another  has  partly  given  it  and  complete 
the  explanation.  This  will  stimulate  attention,  especially  on  the  part 
of  the  unprepared  pupils.  When  by  questioning  or  by  hearing  the 
explanation,  the  backward  pupils  have  learned  how  to  dispose  of  the 
problem  in  hand,  they  should  be  called  upon  to  show  that  they  can 
take  all  the  steps  leading  to  the  correct  solution.  A  good  test  is  to 
change  one  condition  somewhat  and  then  ask  for  a  solution.  Ques- 
tioning, in  most  cases,  is  better  than  telling.  In  questioning  a  pupil 
to  bring  him  to  an  understanding  of  a  problem  or  a  principle,  the 
teacher  must  always  go  back  to  what  is  known,  as  a  starting  point. 
It  is  sometimes  a  good  plan  to  distribute  small  slips  of  paper,  read  a 
problem,  have  each  pupil  write  the  answer  at  a  signal,  after  sufficient 
time  has  elapsed  for  its  solution ;  then  give  another  in  the  same  way 
until  opportunity  to  test  all  with  several  problems  has  been  given. 
Analyses,  explanations,  and  modes  of  solution  given  by  the  pupils 
should  follow. 

(d)  Time  will  be  saved  and  accuracy  of  results  insured  by  encour- 
aging pupils  to  use  the  largest  measures  possible  in  finding  ratios 
or  making  comparisons  between  like  numbers.  Take,  for  instance, 
What  part  of  an  18|-ft.  square  is  a  12i-ft.  square?  If  the  pupil  finds 
that  the  12|-ft.  square  is  twice  as  long  as  a  6^-ft.  square,  and  the 
18|-ft.  square  is  three  times  as  long  as  a  6^-ft.  square,  he  readily  sees 
that  the  ratio  of  the  length  of  the  12|-ft.  square  to  the  length  of 
the  18§-ft.  square  is  as  2  to  3,  or  f.  In  like  manner  he  can  compare 
widths.  After  this  it  is  an  easy  step  to  bring  the  compound  elements 
together,  and  find  that  the  12|-ft.  square  is  $  as  large  as  the  18f-ft. 
square. 

(e)  The  scientific  teacher  will  find  many  other  plans  of  developing 
ideas  of  subjects  presented  in  the  text.  He  should  always  feel  at 
liberty  to  use  his  own  methods  and  devices  if  they  are  consistent 
and  will  not  confuse  past  or  future  lessons.  It  is,  however,  better  to 
know  one  plan  well  than  to  have  a  superficial  knowledge  of  many 
plans. 


CONTENTS 

SECTION'  PAGE 

I.     Introducing  the  Idea  of  Ratio 9 

II.     Introducing   the   Fraction    One  Half  and    developing 

Relations 12 

III.  Introducing  the   Fraction   One    Third  and   developing 

Relations • .         .         .         .18 

IV.  Introducing   the   Fraction    One   Fifth    and    developing 

Relations .         .23 

V.     Introducing  the  Fraction  One  Seventh  and  developing 

Relations 29 

VI.     Developing  the  Idea  of  Decimals  and  their  Relation  to 

Other  Fractions 35 

VII.     Introducing  the  Idea  of  Per  Cent  and  developing  Rela- 
tions          .48 

VIII.     Introducing  General  Compound  Numbers  and  develop- 
ing Relations        .         . 56 

IX.     Introducing  the  Study  of  Surfaces  and  developing  Rela- 
tions       65 

X.     Introducing  the  Study  of  Solids  and  developing  Relations       79 

XI.     Review  of  Measurements 92 

XII.     Introducing  the  Ideas  of  Analysis  and  Mental  Algebra 

in  solving  Problems  and  finding  Relations  .         .         .     101 

XIII.  Reviewing  and  extending  the  Idea  of  Percentage    .         .     122 

XIV.  Reviewing   and   extending  Ideas  previously  presented 

and  establishing  New  Relations  i     134 

7 


SECTION  I 

INTRODUCING  THE  IDEA  OF  RATIO 


1.  Rectangle  a  is  what  part  of  rectangle  b  ? 

2.  Rectangle  b  is  how  many  times  rectangle  a  ? 

3.  What  is  the  ratio  of  rectangle  a  to  rectangle  b  ?    ^. 

4.  What  is  the  ratio  of  rectangle  b  to  rectangle  a  ?    2. 


5.  What  is  the  ratio  of  a  line  1  in.  long  to  a  line  2  in. 
long? 

6.  What  is  the  ratio  of  a  line  2  in.  long  to  aline  1  in. 
long  ? 

7.  What  is  the  ratio  of  a  line  4  in.  long  to  a  line  2  in. 
long  ? 

8.  What  is  the  ratio  of  a  line  2  in.  long  to  a  line  4  in. 
long? 

9.  What  is  the  ratio  of  1  qt.  to  2  qt.  ?    Of  2  qt.  to  1  qt.  ? 

10.   What  is  the  ratio  of  4  qt.  to  2  qt.  ?    Of  2  qt.  to  4  qt.? 

9 


10  THE  IDEA   OF  RATIO 

11.  What  is  the  ratio  of  4  bu.  to  2  bu.  ?  Of  2  bu.  to 
1  bu.  ? 

12.  What  is  the  ratio  of  6  in.  to  3  in.  ?  Of  6  in.  to  2  in.  ? 
Of  2  in.  to  6  in.  ? 

13.  What  is  the  ratio  of  3  gal.  to  9  gal.  ?  Of  9  gal.  to 
3  gal.  ? 

14.  James  has  9  marbles  and  Henry  has  6  marbles. 
Henry's  marbles  equal  what  part  of  James's  marbles? 
What  is  the  ratio  of  Henry's  marbles  to  James's  marbles  ? 
Of  James's  to  Henry's  ? 

15.  A  has  $  12  and  B  has  $  9.  What  is  the  ratio  of  B's 
money  to  A's  ?     Of  A's  to  B's  ? 

16.  What  is  the  ratio  of  1  ft.  to  3  ft.  ?     Of  1  ft.  to  1  yd.  ? 

17.  *  What  is  the  ratio  of  1  yd.  to  1  ft.  ?  Of  1  yd.  to  2  ft.  ? 

18.  What  is  the  ratio  of  1  yd.  to  6  in.  ?     Of  1^  yd.  to  6  in.  ? 

19.  What  is  the  ratio  of  6  in.  to  2  yd.  ?     Of  2  yd.  to  6  in.  ? 

20.  A  boy  measured  a  stick  with  a  6-inch  rule  and 
found  it  to  be  2  yd.  long.  How  many  times  did  he  have 
to  use  the  rule  to  measure  the  stick  ?  The  ratio  of  2  yd. 
to  6  in.  is  what  ? 

21.  What  is  the  ratio  of  1  qt.  to  1  gal  ?  Of  1  qt.  to 
5  gal.  ? 

22.  A. man  is  dipping  water  from  a  barrel  into  a  5-gallon 
can,  using  a  2-quart  dipper.  How  many  times  must  he 
dip  the  dipper  full  to  fill  the  can  ?  The  ratio  of  2  qt.  to 
5  gal.  is  —  ? 

*  The  student  must  observe  that  ratios  can  be  expressed  only  between 
quantities  of  like  name.  What  must  be  done  to  quantities  of  unlike 
names  ? 


THE  IDEA   OF  RATIO  11 

23.  What  part  of  a  canf ill  is  each  dipperful  ? 

24.  At  20  $  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  %  1.20  ?     What  is  the  ratio  of  $1.20  to  20^? 

25.  If  22  yd.  of  cloth  cost  %  15.50,  what  part  of  $  15.50 
will  2  yd.  cost?   4  yd.  ?   5  yd.  ?    9  yd.  ?    11  yd.  ? 

26.  How  many  steps  2  ft.  6  in.  long  must  a  man  take 
to  walk  a  distance  of  10  ft.  ?  What  is  the  ratio  of  120  in. 
to  30  in.  ?     Of  10  ft.  to  1\  ft.  ? 

27.  What  is  the  ratio  of  1  ft.  6  in.  to  1  yd.  1  ft.  ? 

28.  From  a  stick  1  yd.  2  ft.  long,  a  piece  1  ft.  8  in. 
long  was  cut  off.     What  part  of  the  stick  was  cut  off  ? 

29.  From  a  can  containing  3  gal.  3  qt.  of  milk,  1  gal. 
1  qt.  was  sold.     What  part  was  sold  ? 

30.  A  boy  carrying  a  package  of  sugar,  weighing  5  lb. 
8  oz.,  spilled  1  lb.  6  oz.  of  it.  What  part  of  the  sugar  did 
he  spill  ?     What  is  the  ratio  of  1  lb.  6  oz.  to  5  lb.  8  oz.  ? 

31.  What  is  the  ratio  of  5  ft.  to  5  in.  ?  Of  6  ft.  to 
6  in.  ?     Of  12  ft.  to  12  in.  ? 

32.  What  is  the  ratio  of  8  in.  to  1  yd.  1  ft.  ?  Of  1  yd. 
1  ft.  to  8  in.  ? 

33.  What  part  of  a  bushel  is  1  pk.  4  qt.  ? 

34.  What  part  of  1  bu.  2  pk.  2  qt.  of  cherries  is  sold,  if 
1  pk.  2  qt.  is  sold  ? 

35.  -How  many  strips  of  wall  paper  18  in.  wide  will  it 
take  to  cover  a  wall  4  yd.  long  ? 


SECTION  II 

INTRODUCING  THE  FRACTION  ONE  HALF  AND 
DEVELOPING  RELATIONS    . 

SUGGESTIONS 


a.  Pupils  should  think  problems  through  before  attempting  to  give 
expression  to  ratios. 

b.  Analysts  should,  in  the  main,  be  the  oral  expression  of  the 
observations  the  pupils  make  in  thinking  through  their  problems. 
Analyses  should,  if  possible,  be  given  in  short,  simple  sentences  in 
which  the  notion  of  what  each  step  is  should  come  in  its  natural 
order. 

c.  Sure  progress  results  if  pupils  think  relations  in  diagrams. 
When  the  diagram  of  a  class  of  problems  can  be  carried  in  the  mind, 
the  pupils  should  image  it. 

d.  In  presenting  Section  II,  it  is  advisable  to  let  the  diagram  of 
each  new  step  remain  on  the  blackboard  during  the  first  recitation. 
In  subsequent  lessons  it  should  be  reproduced  if  pupils  cannot  recall 
and  image  it.  . 


c 

a 

b 

1.  What  part  of  the  square  is  a  ?     What  part  is  b  +  c  ? 

2.  a  and  b  +  c  together  equal  the  whole  square.     What 


is  the  value  of  \  +  \  ? 


12 


THE  FRACTION   ONE  HALF  13 

3.    What  part  of  b  +  c  is  b  ?     What  part  is  c  ? 

.       4.    What  is  the  ratio  of  b  to  6  +  c  ?     Of  6  to  a  ?     Of  c 
to  a? 

5.  What  part  of  the  whole  square  is  b  ?   What  part  is  <?  ? 

6.  What  part  of  the  whole   square  is  b  +  c  ?     J  -f  ^ 
=  what  ? 

7.  What  part  of  the  whole  square  is  a  +  b  ?     |  +  ? 
=  what  ? 

8.  ^  +  \  +  ^  =  what  ?     What  part  of  the  square  are  a, 
5,  and  c  together  ? 

9.  If  you  take  b  from  the  square,  what  part  remains? 

10.  If  you  take  <?,  what  part  remains  ?     1  —  \  ==  what  ? 

11.  If  you  take  a  from  the  square,  what  part  remains  ? 

12.  If    you   take    b  +  c,    what    part    remains  ?      1  —  j 
=  what  ?     1  —  |  =  what  ? 

13.  What  is  the  ratio  of  J  to  J  ?     The  numbers  must 
have  what  like  name  before  the  ratio  can  be  expressed  ? 

14.  What  is  the  ratio  of  J  to  f  ?     Of  J-  to  f  ?     Off 
to  I?     Of  |  to  |? 

15.  What  is  the  ratio  of  £  to  J  ?     Of  }  to  £  ?     Of  i 
to  f  ?     Of  1  to  1  ? 

16.  What  part  of  the  whole  is  ^  and  ^  of  a  gallon  of 
water  ? 

17.  How  much  more  water  will  it  take  to  fill  a  gallon 
jug  after  a  quart  and  a  pint  have  been  put  in  ? 

18.  What  is  the  ratio  of  2  quarts  and  1  pint  to  1  gal- 
lon ?     Of  1^-  gallons  to  3  pints  ? 


14 


THE  FRACTION  ONE  HALF 


19.    What  is  the  ratio  of  half  a  gallon  to  three  pints  ? 
Of  three  pints  to  one  half  a  gallon  ? 


/ 

e 

a 

d 

b 

20.  What  part  of  the  whole  square  is  d  ?  How  many 
rectangles  like  d  could  be  made  of  the  whole  square  ? 

21.  What  is  the  ratio  of  d  to  b?  Of  /+  e  to  b?  Of 
itol? 

22.  What  is  the  ratio  of  d  to  d  +  b  ?  What  part  of  the 
square  is  d  +  b  ?     What  is  the  ratio  of  \  to  -|  ? 

23.  What  is  the  ratio  of  c?  to  a  ?     Of  $  to  \  ? 

24.  What  is  the  ratio  of  d  to  a  and  f+e?  Of  d  to 
a  +  &  ?     Of  |  to  f  ? 

25.  What  is  the  ratio  of  d  to  a  +  b  +f  +  e?  Of  a  tod? 
Oibtod? 

26.  Of  a  + 6  tod?  Oia+f+etod?  Oia  +  b+f  +  e 
tod? 

27.  If  d  is  taken  from  the  square,  what  part  is  left? 
If  d,  e,  and  /  are  taken  ?  If  b  and  d  are  taken  ?  c?,  5,  /, 
and  e  ?     a  and  d  ?     a  and  b  ?     a,  b,  and  d  ? 

28.  b  —  d  =  what ?     a  —  d? 

29.  What  is  the  ratio  of  £  to  ^?  Tof?  To  J?  To  f? 
To  |?     To  |?     Tof? 


THE  FRACTION   ONE  HALF 


15 


30.    What  is  the  ratio  of  \  to  f ?     To  J?     ToJ?     Tof? 
Tof?     Tof?     To  |?     Tof? 


oi        1  _  1  _  ?        1_1 


-*-» 


8  — 
1—9 


1  =  ? 
4         • 


OO  111-?  Ill 

32.    g+^  —  r     -g--t--2- 


+i 


f  +  i 


1+1=?  f+l=? 

33.  What  part  of  the  whole  square  is  e  ? 

34.  What  is  the  ratio  of  e  to  /?     Of  e,  or  /,  to  d? 
To  b ?     To  a ? 

35.  What  is  the  ratio  of  d  to  each  of  the  other  divisions  ? 
Of  b  to  each?     Of  a  ?     Ofe  +  d?     Ofe  +  5?     Ofe  +  a? 

36.  Using  the  fractions  ^,  1   &,  1  &,  f,  fa  J,  £,  f, 

16'  t'  if'  "8 '  it'  anc^  if?  giye  ^ne  ratio  of  each  to  each  of 
the  others. 

37.  Add  each  one  to  each  of  the  others,  as  -^  +  J,  -^  +  4, 
etc. 

38.  Subtract  each  from  each  of  the  others,  as  f  —  Jg, 

16  -  T6'  \  ~  T6'  etc- 

39.  JofJ  =  ?  J' of  J*  JofJ?  f  of  J?  f  of  i?  |ofi? 


9   1 

a 

d 

b 

40.  What  is  the  ratio  of  g  to  e  ?    What  part  of  the  whole 
square  is  g  ? 

41.  What  is  the  ratio  of  #  to  d  ?     To  e  +  d  ?     To  b  ? 
To  a? 


16  THE  FRACTION  ONE  HALF 

42.  What  is  the  ratio  of  h  to  g?  Of  i  to  g  ?  What 
part  of  the  square  is  h  ?     What  part  is  i  ? 

43.  Give  the  ratio  of  each  part  to  each  of  the  others. 
As  of  i  to  h,  to  g,  to  e,  etc. ;   of  g  to  A,  to  e,  to  d,  etc. 

44.  How  many  64ths  in  1  ?  In  £  ?  In  l  ?  In  }  ?  In 
JL  ?     Tn  -1-  ? 

16  '        ■Ln  32  • 

45.  How  many  32ds  in  1  ?     In  J  ?  J  ?  |  ?  ^  ?  ^  ? 

46.  How  many  64ths  in  |  ? 

Suggestion.     In  \  there  are  £§ ;  in  f  there  are  3  times  ||  or  f  £. 

47.  How  many  64ths  in  f  ?  In  f  ?  In  |  ?  In  T3_  ?  in 
JL?     In-*-?     In-9-?     In  44?     In  4£  ?     In  J4  ? 

16  '        "LU   16  '        A11   16   '        XU    16  '        X11    16   *        1U   16  ' 

48.  How  many  32ds  in  each  of  the  fractions  in  problem 
43? 

49.  A  piece  of  property  is  divided  into  64  equal  shares. 
A  owns  \  of  it,  B  |,  C  -^ .  What  part  of  the  property  do 
all  three  own  ?     How  many  shares  do  all  three  own  ? 

50.  A  pole  32  ft.  long  stands  |  in  the  air,  -fy  in  the 
water,  and  the  rest  in  the  mud.  How  many  feet  of  the 
pole  are  in  the  mud  ? 

51.  Jj  of  a  certain  distance  is  8  miles.  What  is  J  of 
the  distance  ? 

Suggestion.  The  ratio  of  \  to  fc  is  that  of  58f  to  ^?,  or  8.  There- 
fore 8x8  miles,  or  64  miles  =  £  of  the  distance. 

52.  -^2  of  a  certain  distance  is  5  miles.  What  is  ^  of 
the  distance  ?     What  is  the  ratio  of  \  to  ■£%  ? 

53.  T3g  of  a  certain  distance  is  6  miles.  What  is  f  of 
the  distance  ? 


THE  FRACTION   ONE  HALF  17 

54.  If  J  of  a  ton  of  coal  costs  $  3,  how  much  will  §  of  a 
ton  cost  ? 

55.  A  milkman  sold  ^  of  his  milk  to  one  customer,  fa 
of  it  to  another,  and  ^  of  it  to  a  third.  What  part  of  his 
milk  did  he  sell  to  all  three  ? 

56.  A  man  owning  f  1  of  a  piece  of  property  sold  T5g  of 
the  property.     What  part  of  it  did  he  have  left  ? 

57.  A  man  owning  T9g  of  a  piece  of  property  sold  \  of 
what  he  owned.  What  part  of  the  property  did  he  sell  ? 
What  part  did  he  still  own  ? 

58.  Count  by  64ths,  giving  each  fraction  in  its  simplest 
form,  thus:  fa,  fa,  fa,  fa,  etc.,  to  ff.  Note  that  fa  is 
the  difference  .between  adjacent  fractions. 

59.  Count  by  32ds  to  §§.  By  16ths  to  If.  By  8ths 
tof. 

60.  A  lady  bought  some  remnants  consisting  of  §  yd.  of 
velvet,  |  yd.  of  silk,  fa  yd.  of  satin,  and  J  yd.  of  plush. 
How  many  yards  of  cloth  did  she  buy  all  together  ? 

61.  Into  how  many  pieces  fa  of  a  yard  long  can  -|  of  a 
yard  of  ribbon  be  cut  ? 

62.  A  man  having  a  certain  distance  to  ride,  rode  T3g  of 
it  the  first  day,  ^  of  it  the  second  day,  and  ^  °f  ft  ^ne 
third  day.  What  part  of  the  distance  did  he  travel  in  the 
three  days  ?     What  part  had  he  left  to  ride  ? 

63.  A  boy  having  ^  of  a  pie,  divided  it  equally  among 
himself  and  three  other  boys.  What  part  of  a  pie  did 
each  receive  ? 

MCN.  MENT.   AR.  2 


SECTION   III 

INTRODUCING  THE  FRACTION  ONE  THIRD  AND 
DEVELOPING  RELATIONS 


i 

f 

e 

a 

h 

d 

c 

1.  What  divisions  of  the  rectangle  are  equal  ? 

2.  What  part  of  the  rectangle  is  a  ?  b?     What  part  is 
all  besides  a  and  b  ? 

3.  What  is  the  ratio  of  c  to  b  ?     Of  d  to  b  ? 

4.  How  many  parts  the  size  of  c  are  there  in  the  whole 
rectangle  ? 

5.  J  of  1  =  ?     What  is  the  ratio  of  .J  to  \  ? 

6.  What  is  the  ratio  of  e  to  d  ?  *  To  <?  ?    To  5  ?    To  the 
whole  rectangle  ? 

7.  1  of  ^  =  ?     What  is  the  ratio  of  1  to  27  ? 

8.  What  is  the  ratio  of  g,  h,  or  i  to  e  ?     To  /  ?     To  c?  ? 
Toe?     To  6  ?     To  the  whole  rectangle  ? 

9.  J  of  2V  =  ?     What  is  the  ratio  of  fa  to  -2V  ? 

10.    Give  the  ratios  of  each  part  to  each  of  the  other 

parts. 

18 


THE  FRACTION  ONE  THIRD  19 

11.  What  is  the  ratio  of  t  to  /  +  e  ?     To  c  +  d?     To 
a  +  b?     Tof+dt     Tof  +  b?     Tof+a?     Tod  +  b? 

12.  What  is  the  ratio  of  fo  to  £  ?     Of  ^  to  §  ?     Of 
fcto.f?     To^?     To  if?     To|? 

13.  What  part  of  the  rectangle  is  b  +  e?   b  +  c  +  d? 
b  +  c  +  d  +  e? 

14.  1  4- 1  as  ?      IXlss?      1  +  i    .      1    =? 

15.  If  l  is  divided  into  3  equal  parts,  what  is  each  part 
called  ? 

16.  If  i  is  divided  into  3  equal  parts,  what  is  each  part 
called  ? 

17.  If  2V  is  divided  into  3  equal  parts,  what  is  each 
part  called  ? 

18.  How  many  9ths  =  J  ?    How  many  27ths  =  ^  ?    How 
many  81sts  =  ^Y  ? 

19.  -$■  =  how  many  27ths  ?     |  +  ^Y  =  ? 

20.  Express  in  9ths :  1   } ,  f ,  &,  ^  if  if  if  A'  If 

21.  Express  in  27ths  :  f  |,  1  f  f,  f  *  |,  f,  ^_   14,  |£ 

51. 

'22.    Express  in  81sts :  f  f ,  f,  l  f ,  ^  &,  if,  ff . 

23.  If  3ds  are  given,  how  can  you  change  them  to  9ths? 
To  27ths  ?    To  81sts  ? 

24.  If  9ths  are  given,  how  can  you  change  them  to  27ths  ? 
To  81sts  ?     To  3ds  ? 

25.  If  27ths  are  given,  how  can  you  change  them  to 
81sts  ?     To  3ds  ?     To  9ths  ? 

26.  If  81sts  are  given,  how  can  you  change  them  to  3ds  ? 
To9ths?     To27ths? 


20  THE  FR ACTION  ONE  THIRD 

27.  What  is  the  ratio  of  any  number  of  81sts  to  the 
same  number  of  27ths?  To  the  same  number  of  9ths? 
To  the  same  number  of  3ds  ? 

28.  What  is  the  ratio  of  any  number  of  27ths  to  the 
same  number  of  9ths  ?  To  the  same  number  of  3ds  ?  To 
the  same  number  of  81sts  ? 

29.  What  is  the  ratio  of  any  number  of  9ths  to  the 
same  number  of  3ds  ?  To  the  same  number  of  27ths  ?  To 
the  same  number  of  81sts  ? 

30.  What  is  the  ratio  of  any  number  of  3ds  to  the  same 
number  of  9ths  ?  To  the  same  number  of  27ths  ?  To 
the  same  number  of  81sts  ? 

31.  Give  the  ratio  of : 


1    in    1 

-JT  l0  27 

aV t0  i 

i*oj 

iM 

fa t0  aV 

fatoi 

ftof 

ftof 

II  toff 

fa  t0  I 

ftof 

¥t02V 

ifrto* 

*VM 

9  tO  fa 

1  t0  22T 

fat0§ 

aV  to  fa 

4  to  A 

¥  t0  A 

fa t0  i 

w  to  h 

1   fn     1 
9  t0  ^T 

t t0  A 

fa t0  f 

If  to  ff 

t*>* 

¥to|- 

32.  Using  the  fractions  ^  A'  i'  2T>  2T'  t>  2V'  2Y'  h 
add  each  one  to  each  one  following,  as  ^  +  fa  2V  +  9' 

2Y  +  2T>  etc- 

33.  Using  the  same  fractions,  subtract  each  one  from 
each  one  that  follows,  as  27  —  2V  9  —  fa  e^c# 

34.  Add  fa  |,&,  and  3 ' 

35.  What  part  of  9  yd.  is  2  ft.  ? 


THE  FRACTION  ONE  THIRD  21 

36.  If  ^  of  a  rectangle  is  divided  into  2  equal  parts, 
how  many  of  those  parts  are  there  in  the  rectangle  ? 
What  is  £  of  f  ?    i  of  i  =  ?    i  of  1  =  ? 

37.  If  ^  of  a  rectangle  is  divided  into  2  equal  parts, 
how  many  of  those  parts  are  there  in  the  rectangle  ? 
iof|  =  ?    iofi=?    lofi=?     $of*=?     &ofl  =  ? 

38.  If  Jg  of  a  rectangle  is  divided  into  2  equal  parts, 
what   fraction   of  the  whole  rectangle  is  each  of  them  ? 

|of|  =  ?    ^ofl  =  ? 

39.  If  fa  ol  a  rectangle  is  divided  into  2  equal  parts, 
what  fraction  of  the  rectangle  is  each  of  them  ?     1  of  ^  =  ? 

40.  State  the  ways  in  which  1  of  anything  may  be 
found.  Ans.  |  of  |,  |  of  ±,  |  of  1. 

41.  State  the  ways  in  which  fa  of  anything  may  be 
found.     fa  of  anything,     fa  of  anything. 

42.  If  1  of  a  rectangle  is  divided  into  2  equal  parts, 
what  fraction  of  the  whole  is  each  of  them  ?  \  of  -^  =  ? 
lof|  =  ?    lofi  =  ?     lofi  =  ?    TVofl  =  ? 

43.  ^ofJY=?     <rVofi-  =  ?    fral-lml 

44.  State  the  ways  in  which  fa  of  anything  may  be 
found.  fa  of  anything,  fa  of  anything,  fa  of  any- 
thing. 

45.  Count  by  72ds,  thus:  fa,  fa,  fa,  fa,  fa,  fa,  etc., 
to  ft- 

46.  Count  by  54ths  in  a  similar  way.  By  48ths.  By 
36ths.  .  By  24ths.     By  18ths.     By  12ths. 


22  THE  FRACTION  ONE  THIRD 

47.  A  owns  ^  of  a  mine,  B  owns  \  of  it,  C  |,  and  D  -X 
of  it.  What  part  of  the  mine  do  all  four  own  ?  If  E 
owns  the  rest  of  the  mine,  what  part  of  it  does  he  own  ? 

48.  What  is  the  sum  of  -|  and  ^  ? 

49.  Five  boys  bought  a  ball  together.  The  first  con- 
tributed ^  of  the  cost,  the  second  T3^,  the  third  ^,  and  the 
fourth  ^.     What  part  did  the  fifth  boy  contribute  ? 

50.  If  the  ball  cost  48  cts.,  how  much  did  each  give  ? 

51.  A  farmer  sold  |  of  a  load  of  vegetables  to  one 
dealer,  ^  to  another,  ^  to  a  third,  and  the  rest  to  a  fourth. 
What  part  of  the  load  did  the  fourth  buy  ? 

52.  How  many  72ds  in  i|  ?     In  |^  ? 

53.  T5¥  of  a  certain  distance  is  15  miles.  What  is  -£$  of 
the  distance  ?     ^|  of  the  distance  ? 

54.  ^  of  a  certain  distance  is  14  miles.  What  is  the 
whole  distance  ? 

55.  |  of  a  ton  of  coal  costs  $  5.  How  much  does  ^  ton 
cost? 

Suggestion.  The  ratio  of  \  to  f  is  f .  Therefore  \  ton  is  worth 
i  of  $5,  or  $4. 

56.  ^2  °^  a  t°n  of  hay  is  worth  $  10.  How  much  is  § 
of  a  ton  worth  ? 

57.  If  f  of  a  yard  of  cloth  costs  30  ^,  how  much  will  f 
of  a  yard  cost  ? 

58.  T3g  of  a  certain  piece  of  property  is  valued  at  $  3600. 
At  that  rate  how  much  is  f  of  it  worth  ? 

59.  25?  of  a  mining  claim  sold  for  $  2500.  At  the  same 
rate,  for  how  much  should  -^  of  it  sell  ? 

60.  What  is  the  ratio  of  f  to  T%  ?     Of  T^  to  ^  ? 

61.  What  is  the  ratio  of  f  of  f  to  f  of  £  ? 


SECTION  IV 

INTRODUCING  THE  FRACTION  ONE   FIFTH  AND 
DEVELOPING  RELATIONS 


d 

c 

b 

a 

e 

1.  What  part  of  the  rectangle  is  a?  b?  c?  d? 
a  +  b?     a  +  b  +  c?     a  +  b  +  c  +  d? 

2.  What  is  the  ratio  of  e  to  d  ?  Toe?  To  b  ?  To 
a?  Toa  +  b?  Toa  +  b  +  c?  Toa  +  b  +  c  +  d?  What 
part  of  the  whole  rectangle  is  e  ? 

3.  What  is  the  ratio  of  ^  to  i ?  To  *|  ?  To  f  ? 
To  |?     To  f? 

4.  If  a  is  divided  into  2  equal  parts,  what  fraction  of 
the  rectangle  is  each  part  ?     -^  +  yo  =  ? 

5.  What  is  the  ratio  of  TV  to  |  ?      To  ^  ?      To  f  ? 

_3?       4?       5?       10? 
5  '        5  *        5  *        10  * 

6.  How  many  e's  would  it  take  to  equal  J  of  a  ? 

7.  If  5  is  divided  into  3  equal  parts,  what  fraction  of 
the  rectangle  is  each  of  the  parts  ?  Two  of  them  ?  3  ? 
4?     5?     10?     15? 


24  THE  FRACTION  ONE  FIFTH 

8.  |of*=?    |  of*=?    fof|?    fofi?    f  of*? 
^0fi?    jjtofi? 

9.  What  is  the  ratio  of  fa  to  |  ?     To  fa  ?     How  many 
e's  would  it  take  to  make  J  of  cZ  ? 

10.  What  fraction  of  the  rectangle  is  one  of  the  4  equal 
parts  of  c?     2  of  them?     3?     4?     5?     10?     15? 

11.  iofl=?     fofi?     |?     |?     |?     -V>_?     1_5? 

12.  What  is  the  ratio  of  fa  to  |  ?     To  ^  ? 

13.  What  is  the  ratio  of  f  to  fa  ?     f  to  ^  ?     f  to  fa  ? 

14.  What  is  the  ratio  of  f  to  fa  ?  f  to  -^  ?  Of  any 
number  of  5ths  to  the  same  number  of  lOths  ? 

15.  What  part  of  the  rectangle  is  one  of  the  6  equal 
parts  of  a  ?  One  of  the  8  equal  parts  of  a  ?  One  of  the 
9  equal  parts?     10?     12?     15?     16?     18?     20? 

16.  Give  the  ratio  of  each  of  the  following  fractions  to 
each  of  the  others:  \,  fa,  fa  fa  fa,  fa  fa  ^ 

17.  When  the  numerators  of  two  fractions  are  alike, 
what  simple  way  do  you  see  of  telling  the  ratio  of  one  to 
the  other  ?     The  ratio  of  f  to  f  is  f ;  of  f  to  fa  is  ^  or  4. 

18.  Give  the  ratio  of : 


t  to  A 

|to^ 

tw  to  2% 

4*  to  if 

*to^ 

A-  to  -& 

A  to   ifo 

A  to  A 

fto  «& 

tV  to   gV 

■is  to  ^y 

TOT  to  TV 

¥*°H 

«to& 

A  to  & 

*  to  £ 

I  to  ^ 

A  to  a 

<&  to  ^jr 

&t°A 

19.    Give  the  ratio  of  each  of  the  following  fractions  to 
each  of  the  others:  fa  fa  fa  fa  fa  fa  fa  £|. 


THE  FRACTION  ONE  FIFTH  25 

20.  What  is  the  ratio  of  f  to  f  ?  Of  T\  to  T%  ?  Of  f 
to  |? 

21.  When  the  denominators  are  the  same,  what  simple 
way  do  you  see  of  rinding  the  ratio  of  one  fraction  to 
another  ?     The  ratio  of  f  to  f  is  -| ;  of  -fa  to  ^  is  $. 

22.  What  must  be  done  before  finding  the  ratio  of 
fractions  whose  denominators  are  unlike  ? 

23.  What  is  the  ratio  of  |  to  £?     Of  |  to  f  ? 

24.  What  is  the  ratio  of  ^  to  -f?     Of  f  to  f  ? 

25.  If  f  of  a  yard  of  cloth  costs  $1,  how  much  will  a 
yard  cost  ? 

Suggestion.  The  ratio  of  1  yd.  to  f  yd.  is  f.  Therefore  1  yd. 
will  cost  f  of  what  £  of  a  yard  costs,  or  £  of  $  1,  or  $  If. 

26.  If  |  of  a  yard  of  cloth  costs  $  1,  what  will  be  the 
cost  of  |-  of  a  yard  ? 

27.  If  |-  of  a  quantity  of  sugar  costs  $-$-,  how  much  will 
|  of  the  quantity  cost  ? 

28.  If  5  lb.  of  sugar  cost  $  J,  find  the  cost  of  7  lb. 

29.  A  can  mow  \  of  an  acre  in  half  a  day,  and  B  \  an 
acre  in  the  same  time.  How  many  acres  could  both 
together  mow  in  half  a  day  ?     In  a  day  ?     In  2  days  ? 

30.  A  can  do  f  of  a  certain  piece  of  work  in  one  day, 
B  |  of  it  in  the  same  time.  How  many  12ths  of  the  work 
can  both  do  in  a  day  ?  In  Jy  of  a  day  ?  If  they  can  do 
-^2-  of  the  work  in  Jy  of  a  day,  how  many  17ths  of  a  day 
will  it  take  them  to  do  ^|,  or  the  whole  work  ? 

31.  A  can  break  ^  of  an  acre  of  new  land  in  a  day,  B  \ 
of  an  acre.  How  long  would  it  take  both  of  them  to  break 
one  acre  ? 


26  THE  FRACTION  ONE  FIFTH 

32.  James  has  -§  as  m'any  marbles  as  John.  If  John 
has  30  marbles,  how  many  has  James  ? 

33.  v  James  has  |-  as  many  marbles  as  John.  If  James 
has  18,  how  many  has  John  ? 

34.  Four  boys  found  a  sum  of  money,  and  divided  it  in 
such  a  way  that  the  first  got  f  of  it,  the  second  fa,  the 
third  ^,  and  the  fourth  the  remainder.  What  part  of  the 
money  did  the  fourth  get  ? 

35.  A  owned  ^  of  a  mill,  and  B  ^1.  Which  owned  the 
greater  amount,  and  how  much  ? 

36.  A  man  set  out  on  a  four  days'  journey.  The  first 
day  he  traveled  Jl  of  the  distance,  the  second  day  |  of  it, 
the  third  day  -fa  of  it.  What  part  of  the  distance  did  he 
have  to  travel  the  fourth  day  ? 

37.  A  owned  -fa  of  a  mill  and  sold  f  of  what  he  owned. 
What  part  of  the  mill  did  he  sell,  and  what  part  did  he 
still  own  ? 

38.  A  owned  fa  of  a  mill  and  sold  ^  of  the  mill  to  B. 
What  part  of  the  mill  did  A  still  own  ?  What  part  of  his 
share  did  he  sell  ?  What  is  the  ratio  of  A's  share,  after 
selling,  to  B's  ? 

39.  A  has  60  A.  of  land  and  B  50  A.  A  gives  half  of 
his  land  to  B,  receiving  in  return  half  of  B's.  How  many 
acres  has  each  after  the  exchange  ? 

40.  Five  men  together  own  a  factory.  The  first  owns 
|  of  it,  the  second  -fa  of  it,  the  third  jfc  of  it,  and  the 
fourth  -fa  of  it.     What  part  does  the  fifth  own  ? 

41.  If  the  share  of  the  fifth  was  $  2000,  what  was  the 
value  of  fa  of  the  factory?  Of£f?  Of  fa?  Of  if?  Of  fa? 
What  was  the  share  of  each  of  the  first  four  men  ? 


THE  FRACTION  ONE  FIFTH  27 

42.  If  the  yearly  profits  of  the  factory  were  $  9600,  what 
should  the  first  man  receive  as  his  share  of  the  profits  ? 
|  of  $9600=  ?     What  should  each  of  the  others  receive? 

43.  A,  B,  and  C  engaged  in  business.  A  furnished  1 
of  the  capital,  B  \  of  it,  and  C  the  remainder.  What  part 
of  the  capital  did  C  furnish  ? 

44.  If  the  whole  capital  was  $  2400,  how  much  money 
did  A  invest?  B?  C? 

45.  If  their  profits  were  $600,  how  much  should  A 
receive?  B  ?  C  ? 

46.  A,  B,  and  C  rent  a  pasture.  A  puts  in  25  sheep, 
B  30  sheep,  and  C  17  sheep.  What  part  of  the  rent 
should  each  pay  ?  If  the  whole  rent  is  $  36,  how  much 
should  each  pay  ? 

What  is  the  cost  for  1  sheep  if  the  cost  for  72  sheep  is  $  36  ? 

47.  If  gV  °f  a  quantity  of  sugar  cost  $  |,  how  much 
will  {^  of  the  same  quantity  cost  ?     What  is  the  ratio  of 

48.  I  of  a  yard  of  cloth  cost  $  f .  How  much  will  -^ 
of  a  yard  cost  ? 

49.  At  $f  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $  I  ?     What  is  the  ratio  of  $  £  to  $  f  ? 

50.  At  $  21  each,  how  many  hats  can  be  bought  for  $  7 J  ? 
What  is  the  ratio  of  $  -1/-  to  $  f  ?     Of  1 7J  to  $  21  ? 

51.  At  $3J  each  how  many  hats  can  be  bought  for 
$131?    What  is  the  ratio  of  $4f  to  $-!?•?    Of  $131  to  $31? 

52.  How  many  yards  of  ribbon  at  $£-%  each  can  be 
bought  for  $-^?     What  is  the  ratio  of  $-fo  to  $£%? 


28  THE  FRACTION  ONE  FIFTH 

53.  If  you  know  the  cost  of  f  of  a  quantity,  how  can 
you  find  the  cost  of  |-|  of  the  quantity?  What  is  the 
ratio  of  JJ  to  f  ? 

54.  Given  the  cost  of  ■£$  of  a  quantity,  how  can  you 
find  the  cost  of  ^  of  the  quantity  ? 

Ans.    By  taking  |^  of  the  cost  of  ^  of  it. 

55.  Give  the  ratio  of : 

\  to  &.  *M-  |  to  A-  itof  &tof 

ftof.  &t°tt-        ftof.  if  to  if-  ftof. 

56.  Given  the  cost  of  \  of  a  quantity,  how  can  you  find 
the  cost  of  T52  of  it  ? 

Given  cost  of  : 

|,  to  find  cost  of  27o*  ^51  to  find  cost  of  \\. 

^,  to  find  cost  of  \.  f ,  to  find  cost  of  f . 

^,  to  find  cost  of  \.  J$,  to  find  cost  of  ||. 

J,  to  find  cost  of  |.  f ,  to  find  cost  of  J. 

57.  If  3  yd.  1  ft.  of  rubber  hose  cost  $ .90,  how  much 
will  1  yd.  cost  ? 

58.  A  merchant  sold  1  bu.  1  pk.  of  timothy  seed  for 
$1.20.  At  that  rate  how  much  should  he  charge  for 
1  pk.  2  qt.  ? 

59.  8  is  the  ratio  of  what  to  1  gal.  1  qt.  ?  8  is  the  ratio 
of  10  gal.  to  what  ? 

60.  The  cost  of  one  chair  is  J  of  the  cost  of  another 
chair.  If  the  first  chair  cost  $10,  how  much  did  the 
second  cost  ?  If  the  second  cost  $  12,  how  much  did  the 
first  cost  ?     What  is  the  ratio  of  10  to  8  ?     Of  15  to  12  ? 

61.  What  is  the  ratio  of  If  to  1J?  To  j ?  To  |? 
Tof?     To  2?     To5f? 


SECTION  V 

INTRODUCING  THE  FRACTION  ONE  SEVENTH  AND 
DEVELOPING  RELATIONS 


a 

i 

i 

!   c  ! 

1.  What  part  of  the  rectangle  is  a?     b?     c? 

2.  What  is  the  ratio  of  a  to  the  whole  rectangle  ?    Of  b  ? 


Qf  el 

3.    What  is  the  ratio  of  \  to  |  ? 


ftSOf?      fto|?      f? 


t?  f?  £? 

4.  If  a  is  divided  into  2  equal  parts,  what  part  of  the 
rectangle  is  each  part  ?  What  part  are  both  parts  ?  3  such 
parts?    4?     5?     6?     8?     10?     13?     14? 

5.  T\  =  how  many  7ths  ?    T\?    T\?    T\?    £f?    if?   jf? 

6.  Using  the  fractions  fo  f  ^,  f,  T\,  f,  J,  f  T94,  f, 
T?'  f »  il'  I"*  giye  ^ne  ra^io  of  each  to  each  of  the  others. 

7.  If  ^  is  divided  into  3  equal  parts,  how  many  such 
parts  are  there  in  £  ?  What  is  the  name  of  each  part  ? 
Of  2  such  parts  ?     3?     5?     19?     21? 

8.  ^T  =  how  many  7ths?     &?    &  ?     if?    ft?     |f? 

2  1  ? 
21  ' 


30  THE  FRACTION  ONE  SEVENTH 

».  iof*=?      |off=?      Joff=?     I  off-? 
*of*  =  ?    *of$  =  ?    Joff  =  ? 

10.  If  ^  is  divided  into  4  equal  parts,  how  many  such 
parts  are  there  in  £  ?  In  1  ?  If  -^  is  divided  into  2  equal 
parts,  how  many  such  parts  are  there  in  ^|  ?     In  1  ? 

ii.  }of}  =  ?  iofTi3  =  ?  fof|  =  ?  JofT%  =  ? 
What  is  the  ratio  of  J  of  any  number  of  7ths  to  \  of  the 
same  number  of  14ths  ? 

12.  What  must  be  done  to  \  to  produce  -^?  How 
many  times  ^  =  |?     f?     f?     f?     £?     f?     J? 

13.  What  must  be  done  to  |  to  produce  -£%?  What 
must  be  done  to  ^  to  produce  ^  ?    ^  of  ^  =  ?    J  of  ^  =  ? 

14.  |  of  ^  =  ?  Is  ^g  larger  or  smaller  than  jfc  ?  Than 
^3  ?     How  many  7ths  =  f  f  ? 

15.  iofi  =  ?  i0f^  =  ?  Jof^  =  ?  ioff-T 
lof^T  =  ?     TVof|=?     iofTV  =  ?     |bf.*-? 

16.  How  many  7ths  in  §£  ?     In  |f  ?     In  ff  ? 

17.  How  many  llths  are  there  in  anything  ?  What  is 
the  ratio  of  2  to  22  ?  \  of  ^  =  ?  \  of  ^  =  ?  \1  \t 
1  ?     l  ?     i  ?     l  ?     _i_  of  l  =  ?     - Ar  of  I  =  ?     -1-  of  I  =  ? 

1?       1?       1?       1?       1? 

z  -     6  •     y  •     8  •     9  • 

18.  If  a  rectangle  is  divided  into  13  equal  parts,  what 
name  is  given  to  one  of  the  parts  ?  To  2  of  them  ?  To  3 
of  them?    4?     7?     12?     13? 

19    1  of  -1-  =  ?     1  of  -X-  =  ?     -1  ?     4  ?     l  ?     i  ? 

20.  What  name  is  given  to  one  of  the  17  equal  parts  of 
a  rectangle  ?     To  one  of  the  19  equal  parts  ?     To  one  of 


TBE  FRACTION  ONE  SEVENTH  31 

the  23  equal  parts  ?    Of  the  29  equal  parts  ?    31?    37?   41? 
43?  47?  53?  59?  61?  67?  71?  73?   79?   83?   89?   97? 

21.  We  may  think  of  ^  as  being  £  of  -|-,  or  |  of  J,  or  J 
of  ^,  or  \  of  J.  Why  is  it  not  possible  to  think  of  any  of 
the  fractions  indicated  in  problem  20  in  a  similar  way  ? 

22.  i  of  TL  =  ?  1  of  Jg  =  ?  I  of  Jg  =  ?  What  are 
the  values  of  the  similar  parts  of  Txy,  ^,  and  ^9  ? 

23.  If  T3y  of  a  quantity  costs  $  25,  how  much  will  ^9T  of  it 
cost? 

24.  What  is  the  ratio  of  ^  to  ^  ?  Of  ^  to  &  ?  0f 
A  to  ,V     Of  A  to  W?     Of  .A  to  &?     Of.fr  to  &? 

Of  A  to  A? 

25.  y  of  a  certain  distance  is  21  miles.  What  is  ^  of 
the  distance  ? 

26.  A  owned  T5T  of  a  mill,  B  fo  C  ^,  and  D  the  rest. 
What  is  the  ratio  of  D's  share  to  that  of  each  of  the  others  ? 

27.  If  D's  share  is  $1000,  what  is  the  value  of  A's 
share  ?     B's  ?     C's  ?     Of  the  whole  mill  ? 

28.  A  piece  of  property  is  divided  into  63  equal  shares. 
A  owns  y  of  it,  B  ^,  C  £,  and  D  the  remainder.  How 
many  shares  does  each  own?  What  is  the  ratio  of  A's 
part  to  each  of  the  others  ?     Of  B's  ?     Of  C's  ?     Of  D's  ? 

29.  A  pole  77  ft.  long  stands  ^  in  the  mud,  y  in  the 
water,  and  the  rest  in  the  air.  How  many  feet  of  the 
pole  are  in  the  air  ? 

30.  A  miner  found  5  nuggets  of  gold.  The  first 
weighed  y  oz.,  the  second  §  oz.,  the  third  §  oz.,  the  fourth 
^j  oz.,  and  the  fifth  1  oz.     What  was  the  weight  of  all? 

31.  A  man  owning  ^  of  a  piece  of  property  sold  -^  of 
the  property.     What  part  of  the  property  had  he  left  ? 


32  THE  FRACTION   ONE  SEVENTH 

32.  A  man  owning  f-  of  a  mill  sold  |  of  what  he  owned. 
What  part  of  the  mill  did  he  sell  ?  What  part  did  he  still 
own? 

33.  A  can  mow  f  of  an  acre  in  half  a  day,  and  B  |  of 
an  acre  in  the  same  time.  How  many  acres  can  both  mow 
in  half  a  day  ?     In  10  days  ?     In  16  days  ? 

34.  A  can  plow  \  acres  in  one  day,  and  B  \  acres. 
How  long  would  it  take  both  of  them  working  together 
to  plow  1  acre  ?     9  acres  ?     21  acres  ? 

35.  A  can  do  a  certain  piece  of  work  in  7  days,  B  in 
5  days.  What  part  of  the  work  can  A  do  in  1  day  ?  B  ? 
Both  ?  How  many  days  will  it  take  both  to  do  the  work 
if  they  can  do  ^-|  in  1  day  ? 

36.  A  can  plow  a  certain  field  in  4  days,  B  in  5  days. 
How  many  days  would  it  take  both  ? 

37.  A  can  saw  seven  cords  of  wood  in  3  days,  B  in  4 
days,  and  C  in  5  days.  How  many  days  would  it  take  all 
three  to  saw  seven  cords  ? 

38.  James  has  T6T  as  many  marbles  as  John.  If  John  has 
55  marbles,  how  many  has  James  ? 

39.  James  has  ^T  as  many  marbles  as  John.  How  many 
has  John,  if  James  has  42  ? 

40.  Four  boys  found  a  sum  of  money,  and  divided  it  in 
such  a  way  that  the  first  got  f  of  it,  the  second  T3T,  the 
third  f£,  and  the  fourth  the  remainder.  What  part  of 
the  money  did  the  fourth  get  ?  What  is  the  ratio  of  the 
share  of  the  first  to  each  of  the  others  ?  Of  the  second  ? 
Third?     Fourth? 


THE  FRACTION  ONE  SEVENTH  33 

41.  If  the  sum  of  money  found  was  $2.31,  how  many 
cents  did  each  receive  ?  What  is  the  ratio  of  the  share  of 
each  to  the  share  of  each  of  the  others  ? 

42.  A  owned  T5T  of  a  mill,  and  B  \  of  it.  Which  owned 
the  greater  amount,  and  how  much  ? 

43.  A  owned  ^  of  a  mill  and  sold  \  of  his  share.  What 
part  of  the  mill  did  he  still  own  ?  -  What  is  the  ratio  of 
the  part  he  owned  after  selling  to  the  part  he  owned  at 
first  ? 

44.  A  owned  -Jf  of  a  mill  and  sold  \  of  the  mill  to  B. 
What  part  of  the  mill  did  A  still  own  ? 

45.  A  has  60  acres  of  land,  and  B  50  acres  ?  A  exchanges 
1  of  his  land  for  \  of  B's.  How  many  acres  has  each  after 
the  exchange  ? 

46.  The  ratio  of  the  circumference  of  a  circle  to  its 
diameter  is  about  3r2-.*  What  is  the  circumference  of  a 
circle  whose  diameter  is  7  inches  ? 

47.  A  wagon  wheel  is  35  inches  in  diameter.  What  is 
its  circumference  ? 

48.  How  many  times  will  a  wheelbarrow  wheel  14  in. 
in  diameter  turn  in  going  12  ft.  ? 

49.  A  bicycle  wheel  turns  3  times  in  going  22  ft.  What 
is  its  diameter  ? 

50.  What  is  the  ratio  of  1\  ft.  to  28  in.  ? 

51.  A  horse  is  tied  to  a  stake  with  a  rope  14  yd.  long. 
If  he  walks  around  the  stake  in  as  large  a  circle  as  the 
rope  will  permit,  how  many  yards  does  he  travel  in  going 
around  once? 

*  This  ratio  is  a  little  too  large,  but  is  close  enough  for  small  circles. 

BtCN.  MENT.   AR.  3 


34  THE  FRACTION  ONE  SEVENTH 

52.  What  is  the  ratio  of  5  to  7  ?  Of  £  of  5  to  }  of  7  ? 
Of  J  of  5  to  J  of  7  ?  Of  3  x  5  to  3  x  7  ?  Of  \2-  x  5  to 
-^  x  7  ? 

53.  What  is  the  ratio  of  the  circumferences  of  two  circles 
whose  diameters  are  5  and  7  ? 

54.  What  is  the  ratio  of  the  circumferences  of  two  circles 
whose  radii  are  2  J  and  3 J  ? 

55.  A,  B,  and  C  rent  a  pasture  for  $23-|-.  A  puts  in 
20  sheep,  B  21,  and  C  6.  How  much  should  each  pay  ? 
What  is  the  ratio  of  what  A  pays  to  what  B  pays?  To 
what  C  pays  ?  Of  B's  share  to  A's  ?  To  C's  ?  Of  C's 
share  to  B's?  Of  the  number  of  sheep  of  each  to  the 
number  of  each  of  the  others  ?  Of  the  fraction  of  the 
whole  number  of  sheep  each  has  to  the  fraction  of 
the  whole  number  each  of  the  others  has  ? 

56.  A,  B,  C,  and  D  engage  in  business.  A  puts  in 
$1000,  B  $2000,  C  $3000,  and  D  $4000.  At  the  end  of 
a  year  they  find  they  have  gained  $  5000.  What  is  each 
one's  share  of  the  gain  ? 

57.  What  is  the  ratio  of  the  gain  of  each  to  the  gain  of 
each  of  the  others  ?  Of  the  investment  of  each  to  the 
investment  of  each  of  the  others? 

58.  If  you  know  the  cost  of  -^  of  a  quantity,  how  can 
you  find  the  cost  of  -j1^  of  the  same  quantity  ? 

59.  If  -^  of  a  quantity  of  sugar  is  worth  $  35,  what  is 
the  value  of  ^  of  the  quantity  ? 

60.  The  cost  of  a  horse  was  -£  of  the  cost  of  a  cow.  If 
the  horse  cost  $140,  how  much  did  the  cow  cost?  If 
the  cow  had  cost  $  39,  how  much  would  the  horse  have 
cost? 


SECTION  VI 

DEVELOPING  THE  IDEA  OF  DECIMALS  AND  THEIR 
RELATION  TO  OTHER  FRACTIONS 


1.  What  part  of  the  whole  square  is  each  of  the  small 
squares?  How  many  small  squares  are  there  iij  the 
whole  ? 

2.  What  part  of  the  whole  square  are  fifty  of  the  small 
squares  ?  -j5^-  =  what  ?  ^  =  how  many  hundredths  ?  .5?) 
of  the  square  is  the  same  as  what  part  of  the  square  ? 
(Note  the  decimal  point  and  learn  its  use.) 

3.  James  has  24  marbles;  John  has  .50  as  many.  How 
many  has  John  ? 

4.  How  many  small  squares  are  there  in  ^  of  the  large 
square?     In  f  ?  |?  f?  £? 

35 


36  THE  IDEA    OF  DECIMALS 

5.  Give  in  fourths  the  values  of  .75,  .25,  .50,  1.00. 

6.  If  \  —  .25,  how  many  hundredths  are  there  in  J? 
How  many  small  squares  =  \  of  the  large  square  ?  \  ?  f  ? 

3?     1?     4?     5?    3?    6?    1?    8.  ?    4  ?    2  ? 
8  *     2  '     8"  •      8  *     4  '     8  *     8  '     8  *     4  '     2  ' 

7.  Give  in  eighths  the  values  of  .25;  .371;  .121 ;  .621; 
.87J;   .50;   .75;   1.00. 

8.  If  .12|  =  1,  how  many  hundredths  =3^?  ^'?  -|? 
a  ?  _4_  ?  1  ?  X  ?  -6-  ?  &  ?  A  ?  -8-  ?   i  ?  -9-  ?  44  ?  4  ?  44  ? 

16-     16'     ^*     16-     16*     8*     16*      16"      2'      16'      16'     tf"      16' 
12?    3.?    13?    14?    1?    15?    16?     8?    4? 
16  *     4  *     16  •     16  •     8  *     16  '     16  "     8  '     4  " 

9.  Give  the  values  in  sixteenths  of  .31  J;  .50;  .121 ; 
.87|;  .43|;  .18|;  .93|;  '.56J;  .68|;  .75;  .06J;  .37 J; 
.621;  .811;  .25. 

10.    Give  the  ratio  of  : 

.121  to  .871  .871  to  .25  .061  to  .621 

.50    to.43|  .93fto.75  .25    to  .371 

.18|  to  .561  .93|  to  .68|  .25    to  .81| 

.75    to  .371  .75    to  .061  .43f  to  .18f 

.811  to  .25  .621-  to  .81|  .433  to  .371 

.50    to  .25  .871  to  .93|  .81 J  to  .93f 

.75    to  .25  .68f  to  .75  .68}  to  .56% 

«  11.  If  you  know  the  cost  of  .81  \  of  a  quantity,  how  can 
you  find  the  cost  of  .871  of  it  ?  Of  .93f  of  it  ?  Of  .25  ? 
Of  .68f  ?     Of  .621  ?     of  .121  ?     Of  .18|  ? 

12.  If  you  know  the  weight  of  .184  of  a  bag  of  grain, 
how  can  you  find  the  weight  of  .56^  of  it  ?  Of  .43 J  of  it  ? 
Of  .871- of  it?     Of  .50?     Of  .621?     Of  .371? 

13.  If  .25  of  a  quantity  of  coal  weighs  3  T.,  how  much 
does  .75  of  it  weigh?  .50?  .18$?  .621? 


THE  IDEA    OF  DECIMALS  37 

14.  At  the  rate  of  $  .30  for  .371  yd.  of  cloth,  how  much 
will  .871  of  a  yd.  cost?  .62*  ?  .121? 

15.  At  the  rate  of  $.21  for  .43|  of  a  bushel  of  wheat, 
what  is  the  value  of  ft  of  a  bushel?    T%?    11?    If?    J? 

4  •     2  * 

16.  T5g  of  a  piece  of  land  is  worth  $  5000.  What  is  the 
value  of  .43f  of  it  ?  .75  ?  .871  ?  .68f  ?  >QQ  ? 

17.  A  farmer  sold  .18|  of  a  load  of  corn.  What  was 
the  ratio  of  what  remained  to  what  was  sold  ?  Of  what 
remained  to  the  whole  load  ? 

18.  .37 -|  of  a  certain  distance  is  24  miles.  What  is  .87| 
of  it  ?  .43f  of  it  ?  .561  of  {t  ?     The  whole  distance  ? 

19.  James  has  .31^  as  many  marbles  as  John.  If  James 
has  25,  how  many  has  John  ?  If  John  has  32,  how  many 
has  James?  What  is  the  ratio  of  James's  marbles  to 
John's  ? 

20.  The  ratio  of  a  certain  number  to  another  is  .68|. 
The  larger  number  is  96.  What  is  the  other  ?  If  the 
smaller  number  had  been  44,  what  would  the  larger  have 
been  ? 

21.  A  owned  f  of  a  mill,  B  .18f,  C  ft,  D  .12J,  and  E 
the  remainder.  What  part  of  the  mill  did  E  own?  What 
was  the  ratio  of  A's  share  to  each  of  the  others  ?  Of  B's  ? 
OfC's?     OfD's?     OfE's? 

22.  At  $.121  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $.15?     For  $  1  ?     For  $  J-  ?     For  $  11  ? 

23.  At  $6J  each,  how  many  plows  can  be  bought  for 
|  of  $  100  ?     For  T%  of  $  100  ?     For  $  43|  ? 


38  THE  IDEA   OF  DECIMALS 

24.  At  $ .  31 J  each,  how  many  books  can  be  bought  for 
15?     For  $10? 

25.  How  many  quarts  are  there  in  .43|  of  a  bushel? 
What  is  the  ratio  of  .18f  pk.  to  .31£  bu.  ? 

26.  How  many  hundredths  are  there  in  1  ?  In  1  of  1  ? 
Inf?     In*  of*?     In  J?     In^ofJ?     Inl?|?J?i? 

3?     2?     4?     5?    &? 
6  *      3  '      6   '      6  '      6  * 

27.  Give  in  sixths  the  values  of  .331  .50,  .66},  .831 
.16|,  1.00.  Which  of  these  can  be  expressed  exactly  in 
thirds  ? 

28.  Give  the  ratio  of : 

.16}  to  .  50  .  331  to  .  66|  .  66|  to  .  83J 

.50    to  .331  .83£  to  .16|  1.00    to  .831 

.66}  to  .50  .331  to  .50 

.83J  to  .66f  .33J  to  1.00 

29.  If  you  know  the  cost  of  .  33 J  of  a  quantity,  how  can 
you  find  the  cost  of  .66f  of  it?  Of  .50  of  it?  Of  all 
of  it? 

30.  Given  the  weight  of  .83^  of  a  load  of  corn,  how 
can  you  find  the  weight  of  .16f  of  it?  Of  .66f  ?  Of 
.331?     Of  .50? 

31.  How  many  hundredths  are  there  in  ^  ?     In  1  of  \  ? 

In  JW  ?     In  A  ?   1  ?   -3-  ?   l  ?   -4-  ?   1  ?   -5-  ?   A  ?  l ?  J-  ? 
xii  12  .     ±ii  12  .    6  .    12  .    1  .    12  .    3  .    12  .    12  .    2  .    12  . 

_a^  ?     2  ?     _9_  ?     3  ?     10?     6  ?     11?     12?     How  manv  new 
12  *      3  •       12  •     If  •      12  *      6  *      12  *      12  '      X1UW  many  «ew 

decimals  are  there  here  ?     What  are  the  four  ? 

32.  Give  in  twelfths  the  values  of  .91f,  .41f,  .581  .081 
Give  in  twelfths  and  also  in  lower  terms  the  equivalents 
of  .831,  .66},  .50,  .331    .75,  .25,  .16},  1.00. 


THE  IDEA   OF  DECIMALS  39 

33.  Give  the  ratio  of : 

.91|  to  .41f  .66f  to  .75  .16§  to  f 

.41|to.58|  .831  to  1.00  ll  to  .581 

.081  to  .50  .75    to  .41|  .41|  to  1 

.50    to  .081  .66|  to  .581  1.00    to  .16| 

.331  to  .75  .25    to  .75  f  to  .41f 

.16fto.08i  .331  to  .25  .25    to  .91§ 
.331  to  .50                             .121  to  .16| 

34.  If  you  know  the  cost  of  .41|  hundredths  of  a  quan- 
tity, how  can  you  find  the  cost  of  1J  of  it  ?  Of  .75  of  it  ? 
Of  |  of  it?     Of  .621  of  it? 

35.  If  you  know  the  weight  of  .58^  of  a  bag  of  sugar, 
how  can  you  find  the  weight  of  .41|  of  it  ?  Of  .66%  of  it? 
Of  ll  of  it? 

36.  At  the  rate  of  *  .22  for  .91f  yd.  of  cloth,  how  much 
will  1.831  yd.  cost  ?     .50  yd.  ?     .331  vd.  ?     .58J  yd.  ? 

37.  ^  of  a  farm  is  worth  $  2100.     Wha*t  is  the  value  of 


.581  of  it?     Of  .50  of  it? 

Of 

.621  of  it? 

Of  .66|-  of  it  ? 

38.   If  |  of  a  piece  of 

property  is  worth  $62.50,  how 

much  is  the  whole  worth ' 

?     .31 

ri?     .66|? 

.43|?     .581? 

.831?     .93|?     J_?     3? 

ll? 

16  ' 

ll?    a? 

12  •        6  * 

39.    A  owns  y1^  of  a  factory. 

,  B  i  G  1  D  i,  and  E  i. 

What  part  of  the  whole  do  all  own  ? 

40.    Give  the  ratio  of : 

.50    to^         .331  to  | 

i  to  .75 

TV  to  -62i 

.81^to^          .831  to  1 

I  to  .91| 

i  to  .66f 

.93ftof          .50    tof 

f  to  .50 

if  to  .68| 

.43f  tof          .66|toT^ 

H  to  .93f 

I  to  .91| 

.68|toT5e         .75    to-fs 

|  to  .43| 

|  to  .581 

40  THE  IDEA   OF  DECIMALS 

41.  How  many  hundredths  are  there  in  1  ?  In  ^  of  1  ? 
|  ?  |  ?  f  ?  f?  Give  the  value  in  5ths  of  .20,  .60,  .40, 
.80,  1.00. 

42.  How  many  hundredths  are  there  in  ^  ?     In  ^  of  \  ? 

Tn  JL  ?       _2_  ?  1  ?       _3__  ?       _4_  ?        2.  ?       _5_  ?        1  ?        _fi_  ?        3  ? 

,in    10  •        10-  5"         10-        10*        5*        10'         2'         10'         5' 

JLV      JL?       4?  _9_?       10.9. 

10  '        10  '        5  *  10  •        10  ' 

43.  How  many  hundredths  are  there  in  |-  of  -j1^-?  In  ^? 
Tn  __2_  y       l  ?      _s_  ?      1  ?      I  ?      _3_  v      J7_  y      2  ?      _9_  y      i  ? 

ia20'         10-        20-        5-        4*         10-         20-         5'         20'        2* 

11  y     1?     13  y     jL_y     3y     i?     ixy     jl?     loy     20  y 

20*        5'        20*        10"        4"        5'        20"        10'        20*        20  '  - 

44.  Give  in  hundredths  the  values  of  -§,  ^,  J^,  ^,  2%, 

1_8_      16        5        15      J3_     14      12. 
20'    20'    20>    20'    20'    20'    20* 

45.  Give  the  ratio  of  : 

.65  to  .25           .50  to  .55  £  to  .85  if  to  .35 

.40  to  .65          .75  to  .85  f  to  .55  -§-  to  .25 

.70  to  .35          .75  to  .80  |  to  .65  ^  to  .95 

.45  to  .20          .90  to  .15  -^  to  .45  ^  to  .50 

.85  to  .45          :95  to  .60  |#  to  .75  J  to  .75 

46.  If  you  know  the  cost  of  ^  of  a  quantity,  how  can 
you  find  the  cost  of  .25  of  it  ? 

Given  cost  of  -|,  to  find  cost  of  .65. 
Given  cost  of  .85,  to  find  cost  of  -^ 
Given  cost  of  .90,  to  find  cost  of  -^-q. 
Given  cost  of  J-J,  to  find  cost  of  -|. 
Given  cost  of  .75,  to  find  cost  of  .85. 


47.    How  many  hundredths 

;  are  there  in  ^  ? 

Iniofi? 

jl  v    jl  y     j_  y    jl  y    j§  ? 

25  *        25  •        25  •        2  5  *        2Z  ' 

ft*  etc. 

48.    Give  the  ratio  of  : 

.24  to  .36            .20  to  .64 

.32  to  .52 

.68  to  .40 

.76  to  .16            .72  to  .48 

.84  to  .92 

.88  to  .44 

.56  to  .88            .60  to  .64 

.88  to  .76 

.36  to  .80 

THE  IDEA   OF  DECIMALS  41 

49.  How  many  hundredths  are  there  in  J?  ^?  \1  ^? 
1  ?  i  ?  JL?  JL  ?  -1-  ?  J-9  -1-? 

6  '     8  •     10  •     12  •     16  *     20  '     25  * 

50.  Give  the  number  of  hundredths  in  f ,  §,  |,  |,  |,  |, 

_2_       2         2_     _2_     _2_ 
10'    12'    16'    20'    25* 

51.  Give  the  number  of  hundredths  in  f,  |,  f,  |,  |, 

10'    12'    16'    27'    2  5* 

52.  Give  the  number  of  hundredths  in  |,  -|,  |,  |,  ^, 

_4_       4       _4_       4  Tn   5      5      5      JL     __5_     _5_     JL     _5_ 

12'     16'    20'    25*       1U   5'    6'    8'    10'    12'    16'    20'    25* 

53.  Give  the  number  of  hundredths  in  |,  |,  t6q,  y6^,  ^, 

_6_     JL        Tn  1    JL    _7_    JL    JL    _7_        Tn  .§-    _8_    __8_    _8__    _8        8 
20'    25*       -Ln   8'    10'   12'   16'  20'   25*       "*  "8"'   10'    12'   16'  20'  25* 

54.  Give  the  number  of  hundredths  in  -^,  -j9^,  T9g,  ^, 
_9         Tn  l£    44    iD.    JLO    io       in  i-l    44    44    44      In  4i 

25*         A11    10'    12'    16'    20'    25*        "L11    12'    16'    20'  25*  -111    12"' 

12      12      12          Tn    13      la     13          In    14      14      14  In  15      15 

16'    20'    25*       Xil    16'    20'    25*        ±1L    16'    20'    25*  AU  16'    20' 

15.           In    15.      l£     16.          Tn    11     11     JL8      11     19  19  2  0     _2  0 
111     16'    2  0'    2  5*         in    2  0'    2  5'    2  0' 


2  5*         X11     16'    2  0'    2  5*         Aii    2  0'    2  5'    2  0'    2  5'    2  0'    2  5'    2  0'    2  5* 
7n     21      22      23      24      25 
111    25'    2  5'    2  5'    2  5'    2^' 

55.*    Give  the  ratio  of : 

.25  to  .16|        .80    to  .66|  .061  to  .081  .60    to  .371 

.04  to  .121      i.oo    to  .18f  .081  to  .10  .621  to  .41| 

.05  to  .331        .go    to  .75  .121  to  .06^  .50    to  .15 

.25  to  .331        .68f  to  M  .66|  to  .50  .68f  to  .91f 

.50  to  .40  .621  to  .831  .121  to  .16f  .40    to  .66| 

56.  A  certain  piece  of  property  is  divided  into  120 
shares.     A  owns  \  of  it,  B  /■§•  of  it,  C  \  of  it,  D  1  of  it,  and 

*  In  solving  such  questions  as  these  the  pupils  will  generally  find  it 
easiest  to  change  the  decimal  form  to  the  corresponding  common  fraction. 
But  if  by  inspection,  as  in  the  ratio  of  .12^  to  .06^,  the  pupils  see  that 
.12|  is  twice  .06£,  it  is  advisable  that  they  shall  go  directly  to  the  point 
and  state  the  fact.     See  also  problem  17,  page  24. 


42  THE  IDEA   OF  DECIMALS 

E  the   remainder.     How  many   shares  does   each   own  ? 
How  many  hundredths  of  the  property  does  each  own  ? 

57.  .18|  of  a  certain  distance  is  7  J  miles.  What  is  .50 
of  the  distance  ?     The  whole  distance  ? 

58.  .55  of  a  certain  distance  is  22  miles.  What  is  .68| 
of  the  distance  ?     .25  of  it  ?     The  whole  distance  ? 

59.  If  .33J  of  a  ton  of  coal  costs  $  2J,  what  is  the  cost 
of  .80  of  a  ton  ?  What  is  the  ratio  of  £  to  Ty?  What  is 
the  value  of  \2- of  $  §? 

60.  Into  how  many  pieces  ,12|  yd.  long  can  2.37|  yd. 
of  ribbon  be  cut  ? 

61.  At  12|^  per  yard,  how  many  yards  of  cloth  can  you 
buy  for  $2. 37 f? 

62.  A  can  mow  .66|  of  an  acre  in  half  a  day  ;  B  .75  of 
an  acre  in  the  same  time.  How  many  acres  can  both  mow 
in  half  a  day  ?     In  a  day  ?     In  12  days  ? 

63.  John  can  do  a  piece  of  work  in  1.5  days  ;  James  in 
1.33 J  days.     How  long  will  it  take  them  both  to  do  it  ? 

64.  Charles  has  .68|  as  many  marbles  as  Henry.  If 
Henry  has  64,  how  many  has  Charles  ? 

65.  Charles  has  .68|  as  many  marbles  as  Henry.  If 
Charles  has  55,  how  many  has  Henry  ? 

66.  .41|  is  the  ratio  of  the  cost  of  one  chair  to  the  cost 
of  another.  If  the  first  cost  $10,  how  much  did  the 
second  cost  ?  If  the  second  cost  $  6,  how  much  did  the 
first  cost  ? 

67.  A,  B,  and  C  engaged  in  business.  A  furnished  .37 J 
of  the  capital,  B  .31^  of  it,  and  C  the  remainder.  What 
part  did  C  furnish  ? 


THE  IDEA   OF  DECIMALS  43 

68.  If  the  whole  capital  was  $3200,  how  much  money 
did  A  invest?     B?     C? 

69.  If  their  profits  were  $1600,  how  much  should  A 
receive  ?     B  ?     C  ? 

70.  What  is  the  ratio  of  A's  share  of  the  profits  to  B's 
share?  A's  to  C's  ?  B's  to  C's?  B's  to  A's?  C's  to  A's? 
C's  to  B's  ? 

71.  What  is  the  ratio  of  A's  share  of  the  capital  to  B's 
share  ?  A's  to  C's  ?  B's  to  C's  ?  B's  to  A's  ?  C's  to  A's  ? 
C's  to  B's  ? 

72.  Given  the  cost  of  .41|  of  a  quantity,  how  can  you 
find  the  cost  of  .58^  of  it  ? 

Given  the  cost  of  .16§,  to  find  cost  of  .12|. 
Given  the  cost  of  .25,    to  find  cost  of  .91§. 
Given  the  cost  of  .75,    to  find  cost  of  .25. 
Given  the  cost  of  .43|,  to  find  cost  of  .50. 
Given  the  cost  of  .87^,  to  find  cost  of  .93|. 
Given  the  cost  of  .18|,  to  find  cost  of  .43|. 
Given  the  cost  of  .81^,  to  find  cost  of  .93 J. 
Given  the  cost  of  .85,    to  find  cost  of  .45. 
Given  the  cost  of  .45,    to  find  cost  of  .20. 

73.  .18|  is  the  ratio  of  12  gal.  to  what  ? 
.66§  is  the  ratio  of    5  ft.  to  what  ? 

1.12J  is  the  ratio  of  18  bu.  to  what? 

.91|  is  the  ratio  of  27 j  lb.  to  what  ? 

1.93|  is  the  ratio  of  62  mi.  to  what  ? 

74.  A  pole  34  ft.  long  stood  .31^  in  the  water,  .33^  in 
the  mud,  and  the  rest  in  the  air.  How  many  feet  were  in 
the  air  ? 


44  THE  IDEA    OF  DECIMALS 

75.  A  pole  stood  .25  in  the  mud,  .41f  in  the  water,  and 
15J  ft.  in  the  air.  How  long  was  the  pole  ?  How  many 
feet  were  in  the  mud  ?     In  the  water  ? 

76.  A  man,  owning  .80  of  an  acre  of  land,  sold  .18 J  of 
his  share  for  $6.     What,  was  the  price  per  acre  ? 

77.  The  value  of  a  certain  man's  property  was  $  6000. 
.81^  of  it  was  real  estate,  .02^  was  cash,  and  the  remainder 
live  stock.     What  was  his  live  stock  worth  ? 

78.  A  man  owning  .40  of  a  piece  of  property  sold  .37^ 
of  his  share.     How  many  hundredths  did  he  still  own  ? 

79.  A  man  chops  2J  cd.  of  wood  in  1.121  days.  How 
long  should  it  take  him  to  chop  20  cd.  ? 

Suggestion.    It  will  take  him  20  x  §  of  f  days. 

80.  What  is  the  cost  of  6  lb.  of  butter,  at  the  rate  of 
$.371  for  fib.? 

81.  .41f  of  72  is  .311  of  what  number? 

82.  50  is  .121  of  .66f  of  what  number? 

83.  If  .  83|  of  a  quantity  of  sugar  is  worth  $  14,  what  is 
the  value  of  .621  0f  the  quantity  ?  Why  is  it  worth  |  of 
$14? 

84.  At  $  .16f  each,  how  much  will  33  note  books  cost  ? 
What  is  the  cost  of  32  note  books  at  $.18f  each ? 

85.  How  many  pounds  of  butter  can  you  buy  for  $1.25 
if  |  lb.  cost$.18f  ? 

86.  How  many  cents  are  there  in$f?  In$|?  In  $  f  ? 
In$T5_?     In  $^?     In$|?     In$f? 

87.  What  part  of  a  dollar  is  25  cents?  331^?  20/? 
18f^?   41f^?   93f^? 


THE  IDEA   OF  DECIMALS  45 

88.  A  owned  .66%  of  a  mill  and  sold  .37 J  of  his  share. 
How  many  hundredths  of  the  mill  did  he  still  own  ? 

89.  A  man  bought  a  horse  and  carriage,  paying  for  the 
latter  .68f  of  what  he  paid  for  the  former.  If  the  carriage 
cost  $  132,  how  much  did  the  horse  cost  ? 

90.  A  man  bought  a  horse  and  carriage,  paying  $192 
for  the  latter.  If  the  ratio  of  the  cost  of  the  carriage  to 
the  cost  of  the  horse  was  .68f,  how  much  did  the  carriage 
cost? 

91.  One  chair  cost  .66%  as  much  as  another.  If  both 
cost  $  50,  what  was  the  cost  of  each  ? 

92.  Ho w.  many  rods  are  there  in  the  circumference  of  a 
circle  .56  rd.  in  diameter? 

93.  What  is  the  ratio  of  .08 J  ft.  to  .56%  in.  ? 

94.  Give  the  ratio  of : 


.16|  to  .25 

.25    to  .66% 

.621- to  .831 

.12J  to  .081 

.75    to  .18} 

.60    to  .371 

.25    to  .331 

.371  to  .25 

.43|  to  .70 

.20    to  .50 

.30    to  .18f 

.75    to  .56% 

.04    to  .16% 

.80    to  .66% 

.68f  to  .91| 

.16f  to  .66$ 

.331  to  .50 

.66f  to  .831 

.40    to  .12£ 

.41f  to.31£ 

.84  to  .561 

95.  -^  of  a  piece  of  land  is  worth  $800.     What  is  the- 

value  of  .43f  of  it? 

96.    A  farmer  sold 

.564  of  a  load  of 

corn.     What  was 

the  ratio  of  what  remained  to  what  was 

sold? 

97.    .25  of  a  certain  distance  is  60  miles.     What  is  .58J 
of  it  ?     .18|  of  it  ?     .371  of  it  ?     .91f  of  it  ? 


46  THE  IDEA   OF  DECIMALS 

98.  James  has  .83^  as  many  marbles  as  John.    If  James 
has  50,  how  many  has  John  ? 

99.  The  ratio  of   a  certain  number  to  another  is  f . 
The  smaller  number  is  .621.     What  is  the  other  ? 

100.  At  $  .18 J  a  yard,  what  part  of  a  yard  can  be  bought 
for  |.16  ?     What  is  the  ratio  of  9&  to  '••£? 

101.  At  $  .41 J  per  yard,  how  many  yards  can  be  bought 
for  $ .  621  ?  For  $ .  31|  ?  For  $  1. 25  ?  What  is  the  ratio 
offto&?     Of^to^?     Offto^? 

102.  At  $.16$  per  yard,  how  many  yards  of  cloth  can 
be  bought  for  $.25?     For*. 33$?    For  J. 12 J?    For  $.40? 

103.  At  $.56;|-  per  pound,  how  many  pounds  can  be 


bought  for  $.75?     For  $  1.12J?     For  $21 


? 
104.    What  is  the  ratio  of  .87$  pk.  to  .24  bu.  ? 


105.  How  many  quarts  are  there  in  .93 J  bu.  ?  In 
.68|  bu.  ? 

106.  If  -y-  of  a  lot  is  worth  $  91$,  what  is  J  of  the  lot 
worth  ?     f  of  it  ?     T%  of  it  ? 

107.  .41 1  of  a  certain  distance  is  80  miles.  What  is 
.  311  of  the  distance  ?     .  91f  of  it  ?     .  58£  of  it  ? 

108.  .68|  of  a  certain  distance  is  99  miles.  What  is 
.91f  of  the  distance  ?     .75  of  it  ?     .93f  of  it  ?     All  of  it  ? 

109.  If  ,66|  of  a  ton  of  coal  is  worth  $2,  what  is  the 
value  of  .50  of  a  ton?  Of  .25  of  a  ton?  Of  .12J  of  a 
ton? 

110.  Into  how  many  pieces  .25  yd.  long  can  10^  ft.  of 
rope  be  cut  ?     101  yd.  of  rope  ? 

ill.  At  $.06^-  each,  how  many  pencils  can  be  bought 
for  $1.25?    For  $1.50?    For  $.75?    For$l? 


THE  IDEA    OF  DECIMALS  47 

112.  At  $  .08  J  each,  how  many  blank  books  can  you 
buy  for  $.75?     For  $.25?     For  $1.50? 

113.  A  can  mow  .75  of  an  acre  in  .33^  of  a  day  ;  B  can 
mow  .66%  of  .an  acre  in  the  same  time.  How  many  acres 
can  both  mow  in  5  days  ?     In  8  days  ? 

114.  Give  in  hundredths  the  values  of :  |,  T8g,  ^,  f,  -^g-, 

11    11     5     4    14    18     4     11    JL    _i>_ 
12?   16'   8"'   5'    16'  2  5'  ?'   2  0'    12'   16* 

115.  The  ratio  of  the  wheat  in  one  bin  to  that  in  another 
is  .831.  If  there  are  60  bu.  in  the  second  bin,  how  many 
are  there  in  the  first  ? 

116.  A,  B,  and  C  engage  in  business.  A  invests  $  2000, 
B  $3000,  and  C  $1000.  How  many  hundredths  of  the 
profits  should  each  receive  ? 

117.  A  farmer  sold  a  cow  for  $  45,  which  was  f  of  what 
he  paid  for  her.  How  many  hundredths  of  the  cost  did 
he  gain  ? 

118.  If  a  merchant  sells  goods  costing  $48,  so  as  to 
gain  .16  J  of  the  cost,  how  much  does  he  gain  ? 

119.  .83|  of  A's  money  equals  .75  of  B's.  What  is  the 
ratio  of  A's  money  to  B's  ? 

120.  .311  of  A's  property  is  land,  .41|  of  it  is  buildings, 
and  the  remainder  is  cash.  If  all  his  property  is  valued  at 
$48,000,  what  is  the  value  of  his  land?  Of  his  build- 
ings ?     How  much  cash  has  he  ? 

121.  A  man  bought  a  horse  and  carriage  for  $  282.  If 
.18|  of  the  cost  of  the  horse  equaled  .40  of  the  cost  of  the 
carriage,  what  was  the  cost  of  each  ? 


SECTION  VII 

INTRODUCING  THE  IDEA  OF  PER  CENT  AND  DEVELOP- 
ING RELATIONS 

1.  What  is  the  ratio  of  1  to  2  ?  Of  2  to  4  ?  Of  50  to 
100  ?  Another  name  for  this  relation  is  50  per  cent,  writ- 
ten 50%,  which  means  the  same  as  .50. 

2.  What  is  50%  of  2  ?  Of  4?  Of  100?  Of  $4? 
Of  $5?     Of  18  bu.?     Of  60  min.? 

3.  |2  is  50%  of  what?  $6  is  50%  of  what?  $10? 
40  lb.  ?     100  bu.  ? 

4.  What  part  of  anything  is  25%  of  it?  (.25  or  J.) 
What  is  25%  of  $12?     Of  4  bu.  ?     Of  100  lb.? 

5.  How  many  per  cent  are  there  in  the  whole  of  any- 
thing?    In  1  of  it?     ^ofit?     |  of  it? 

6.  What  is  75%  of  $20?  50%  of  7  lb.?  25%  of 
16  bu.?     100%  of  $5? 

7.  Give  the  values  in  per  cent  of  the  following:  l,  ^, 

fi    8>    •J-^2'    8'    ,0,2>    ¥'    ,u^2'    8'    'ni2'    8* 

8.  What  is  121%  0f  24?  371%  of  16?  62-|%  of  32? 
87|-%  of  100  ?     75%  of  $1  ?     100%  of  $1  ? 

9.  The  amount  of  grain  in  a  certain  bin  was  80  bu. 
621%  0f  it  Was  sold.  How  many  bushels  were  sold? 
What  per  cent  was  left  ?  30  bu.  is  what  per  cent  of 
80  bu.  ?     50  bu.  is  what  per  cent  of  80  bu.  ? 

48 


THE  IDEA   OF  PER  CENT  49 

10.  A  grocer  sold  50  lb.  of  sugar,  which  was  12^-%  of 
what  he  had  in  stock.  How  much  did  he  have  in  stock  ? 
What  per  cent  did  he  have  left  after  selling  ?  How  many- 
pounds  of  sugar  had  he  left  ?     350  is  what  per  cent  of  400  ? 

11.  Give  the  values  in  per  cent  of  the  following:  ^ 

Also    Of    3,    ^,    g,    6,    T2>    12' 


JL 

JL 

JL 

9 

11 

13      15 
16'     16 

16' 

16' 

16' 

16' 

16' 

•L 

11 

12' 

12* 

12.  How  many  pounds  are  there  in  .16f  of  100  lbs.  ? 

What  is  16|%  of  100  lbs.  ?     What  is  .41f  of  144  bu.  of 

corn  ?     What  is  41|%  of  144  bu.  of  corn  ? 

1 

13.  In  a  tank  there  were  132  gal.  of  water.     58  J  %  of 

it  was  drawn  off.     How  many  gallons  were  drawn  off? 
What  per  cent  of  the  whole  amount  remained  ? 

14.  A  farmer  sold  at  a  gain  of  8|%  a  cow  that  cost  him 
$36.  How  much  did  he  gain?  What  was  his  selling 
price  ? 

15.  A  farmer  sold  a  cow  at  a  gain  of  $  3,  which  was 
8|%  of  what  she  cost.  How  much  did  she  cost?  If  $3 
is  8^%,  or  ^2  of  the  cost,  how  many  dollars  equal  100%, 
or  1|  of  the  cost  ? 

16.  A  farmer  sold  a  cow  for  $  39,  thereby  gaining  8^%. 
How  much  did  she  cost  ?  If  $  39  is  Jf  of  the  cost,  what 
is  the  cost  ? 

17.  A  farmer  sold  for  $39  a  cow  that  cost  him  $36. 
What  per  cent  of  the  cost  did  he  gain  ?  $  3  is  what  part 
of  $  36  ?     What  per  cent  of  $  36  ? 

18.  A  dealer  sold  a  watch  at  a  loss  of  12 J%.  If  it  cost 
him  $48,  how  much  did  he  lose?  What  was  his  selling 
price  ? 

MCN.  MENT.  AR.  — 4 


50  THE  IDEA   OF  PER   CENT 

19.  A  dealer  sold  a  watch  for  $42,  which  was  121%  less 
than  it  cost  him.     What  was  the  cost  ?     The  loss  ? 

20.  A  dealer  sold  a  watch  at  a  loss  of  12|%.  If  his 
loss  was  $ 6,  what  was  the  cost  ?     The  selling  price  ? 

21.  A  watch  that  cost  $48  was  sold  for  $42.  What 
was  the  loss  per  cent  ? 

22.  A  man  having  $4000  invested  20%  of  it  in  corn, 
30%  of  it  in  wheat,  and  the  remainder  in  oats.  How 
much  did  he  invest  in  each  kind  of  grain  ?  What  per 
cent  of  his  money  did  he  invest  in  oats  ? 

23.  A  50-gallon  barrel  is  60%  full  of  oil.  How  many 
gallons  of  oil  are  in  it  ? 

24.  A  has  $1000  invested  in  a  factory,  and  B  $1400. 
At  the  end  of  a  year  they  find  they  have  gained  16|%. 
What  is  each  one's  share  of  the  gain  ? 

25.  A  owned  §  of  a  mill  and  sold  75%  of  his  share. 
What  part  of  the  mill  did  he  sell  ?  What  per  cent  of  the 
mill  did  he  sell  ?     What  per  cent  did  he  still  own  ? 

26.  What  part  of  66  is  22  ?  What  per  cent  of  66  is  22? 
What  per  cent  of  18  is  9  ? 

27.  What  per  cent  of : 


24  is  3  ? 

39     is  13  ? 

25  is  24? 

100  is  80? 

12  is  6? 

30    is  20? 

25  is  25? 

100  is  125? 

18  is  3? 

25    is  20? 

25  is  26  ? 

75  is  125? 

48  is  8? 

41f  is  16|? 

25  is  30  ? 

32  is  28  ? 

50  is  12J? 

28    is  21? 

10  is  10? 

48  is  44? 

28.   A  has  50  A.  of  land,  and  B  60  A.     What  per  cent 
of  A's  land  is  B's  ?     Of  B's  is  A's  ? 


THE  IDEA   OF  PER   CENT  51 

29.  A  has  50  A.,  and  that  is  83  J  %  of  what  B  has.  How 
many  acres  has  B  ?  B  has  60  A.,  which  is  120%  of  what 
A  has.     How  many  has  A  ? 

30.  After  selling  some  cattle  a  dealer  found  that  he  had 
gained  $  30,  which  was  18f  %  of  the  cost.  How  much  did 
they  cost  ?     What  was  his  selling  price  ? 

31.  If  A  has  33^%  more  money  than  B,  what  per  cent 
of  A's  money  is  B's  ?     What  per  cent  of  B's  is  A's  ? 

32.  A  dealer  sold  some  cattle  for  $  190,  thereby  gaining 
18f  %.     How  much  did  they  cost  him  ? 

33.  Goods  were  sold  at  auction  for  $30,  at  a  loss  of 
25%.     How  much  did  they  cost  ? 

34.  16f%  of  a  certain  number  is  2  greater  than  12  J  %  of 
it.     What  is  the  number  ? 

35.  The  difference  between  two  numbers  is  3,  and  one 
of  them  is  25%  greater  than  the  other.  What  are  the 
numbers  ? 

36.  One  number  is  3  less  than  another.  One  is  20% 
less  than  the  other.     What  are  the  numbers  ? 

37.  A  bicycle  was  sold  for  $  50«at  a  loss  of  50%.  How 
much  did  it  cost  ?     What  was  the  loss  ? 

38.  A  newsboy  buys  papers  at  the  rate  of  2  for  3  cents 
and  sells  them  for  2^  apiece.  What  per  cent  does  he 
gain  ? 

39.  If  a  merchant  sells  \  a  pound  of  butter  for  what 
|  of  a  pound  cost,  what  per  cent  does  he  gain  ? 

40:  By  selling  apples  at  the  rate  of  3  for  5  cents,  a  dealer 
gains  25%.  What  per  cent  would  he  have  lost  had  he 
sold  them  at  the  rate  of  5  for  6  cents  ? 


52  THE  IDEA   OF  PER  CENT 

41.  A  farmer  sold  2  cows  for  $24  each.  On  one  he 
gained  20%,  and  on  the  other  he  lost  20%.  How  much 
did  each  cost  ? 

42.  3%  of  a  certain  number  is  2  less  than  5%  of  the 
number.     What  is  the  number  ? 

43.  A  clothing  dealer  sells  a  suit  of  clothes  marked  $  20, 
at  25%  less  than  the  marked  price  and  still  gains  50%  on 
the  cost.     What  is  the  cost  ? 

44.  8%  of  a  certain  sum  is  $30  less  than  14^%  of  it. 
What  is  the  sum  ? 

45.  A  boy  buys  oranges  at  the  rate  of  4  for  10  cts.,  and 
sells  them  at  the  rate  of  3  for  10  cts.  What  is  his  per 
cent  of  gain  ? 

46.  If  3  oranges  are  sold  for  what  4  oranges  cost,  what 
is  the  gain  per  cent  ? 

47.  What  is  the  loss  per  cent  if  4  oranges  are  sold  for 
what  3  cost  ?     5  for  what  4  cost  ? 

48.  What  is  the  gain  per  cent  if  3  newspapers  are  sold 
for  what  5  cost  ?     If  3  are  sold  for  ^  of  what  5  cost? 

49.  A  grocer  buys  strawberries  at  the  rate  of  5  qt.  for 
12  cts.,  and  sells  them  at  the  rate  of  3  qt.  for  12  cts.  What 
is  his  gain  per  cent  ?  How  much  does  he  make  on  30  qts. 
of  berries  ? 

50.  A  bicycle  is  marked  $  48,  but  in  order  to  make  a 
sale  the  dealer  has  to  allow  a  discount  of  12 J%.  How 
much  did  the  bicycle  cost  him,  if  he  made  16|%  on  the 
transaction  ? 

51.  If  a  coal  dealer  sold  f  of  a  ton  of  coal  for  what  f  of 
a  ton  cost,  what  per  cent  did  he  gain  ? 


THE  IDEA   OF  PER  CENT  53 

52.  An  overcoat  is  sold  for  $  24  at  a  gain  of  20% .  How 
much  did  it  cost  ? 

53.  If  a  horse  that  cost  $  174  was  sold  for  $  203,  what 
per  cent  was  gained  ? 

54.  What  is  12|%  0f  16|^  0f  95? 

55.  20%  of  50%  of  a  certain  number  is  78.  What  is 
the  number  ? 

56.  5  times  25%  of  a  certain  number  is  375.  What  is 
the  number  ? 

57.  24  is  12  times  18|%  of  a  certain  number.  What  is 
the  number  ? 

58.  12 1  %  of  a  certain  number  plus  16|%  of  it  equals 
21.     What  is  the  number  ? 

59.  What  number  is  it  of  which  25%  exceeds  18f  %  by  20  ? 

60.  31|%  of  one  number  equals  41|%  of  another.  If 
the  first  is  48,  what  is  the  second  ? 

61.  62  J  %  of  one  number  equals  83^%  of  another.  The 
sum  of  the  two  numbers  is  56.     What  are  the  numbers  ? 

If  f  of  the  first  number  equals  §  of  the  second,  what  is  the  ratio  of 
f  of  the  first  number  to  the  second  ?  Since  the  first  number  is  §  of  f, 
or  f  of  the  second,  both  numbers  are  how  many  thirds  of  the  second 
number  ?  If  56  =  |  of  the  second  number,  what  is  the  second  number  ? 
The  first? 

62.  37^-%  of  James's  money  equals  30%  of  John's.  If 
both  have  81  cts.,  how  much  has  each  ? 

63.  A  certain  part  of  a  pole  37  ft.  long  is  in  the  ground, 
and  the  rest  in  the  air.  18 j  %  of  the  part  in  the  air  equals 
in  length  58J%  of  the  part  in  the  ground.  What  is  the 
length  of  each  part  ? 


54  THE  ILEA   OF  PER   CENT 

64.  A  watch  that  cost  $65  was  sold  for  $52.  What 
was  the  loss  per  cent  ? 

65.  A  man  having  $  60,000,  invested  20  <fc  of  it  in  cot- 
ton, 25  ^  of  it  in  sugar,  15  ^>  of  it  in  molasses,  and  the 
remainder  in  corn.  How  much  corn  did  he  buy,  if  he 
paid  $.30  per  bushel? 

66.  A  has  $2000  invested  in  a  factory,  B  $2400,  and 
C  $1600.  At  the  end  of  a  year  they  find  they  have 
gained  25%  on  the  whole  investment.  What  is  each  one's 
share  of  the  gain  ? 

67.  A  owned  f  of  a  mill  and  sold  50%  of  his  share  for 
$  3000.  What  was  the  value  of  the  mill  at  that  rate,  and 
what  was  the  value  of  A's  share,  both  before  and  after 
selling  ? 

68.  A  has  |  as  much  land  as  B.  What  per  cent  of  A's 
land  is  B's  ? 

69.  If  A  has  25%  more  money  than  B,  what  per  cent  of 
A's  -money  is  B's  ?     What  per  cent  of  B's  is  A's  ? 

70.  Goods  were  sold  at  auction  for  $  62J,  which  was  at 
a  loss  of  37^  %.     How  much  did  they  cost  ? 

71.  The  difference  between  two  numbers  is  5,  and  one 
of  them  is  33^%  greater  than  the  other.  What  are  the 
numbers  ? 

72.  By  selling  at  the  rate  of  4  for  5  cents  what  was 
bought  at  the  rate  of  5  for  6  cents,  what  per  cent  is  gained  ? 

73.  A  farmer  sold  two  horses  for  $  75  each.  On  one  he 
gained  50%,  and  on  the  other  he  lost  50%.  What  was 
the  cost  of  each  ? 


THE  IDEA    OF  PER   CENT  55 

74.  A  couch  is  marked  $35,  but  in  order  to  make  a 
sale,  the  dealer  gives  a  discount  of  20%.  If  he  still  gains 
12%,  what  is  the  cost? 

75.  If  a  grocer  sells  |  of  a  dozen  eggs  for  what  %  of  a 
dozen  cost,  what  per  cent  does  he  gain  ? 

76.  What  is  18f  %  of  133£%  of  84  ? 

77.  27  is  75%  of  75%  of  what  number? 

78.  60  is  6  x  31|%  0f  what  number  ? 

79.  28  is  331%  +  25%  of  what  number? 

80.  What  number  is  it  of  which  66f  %  exceeds  5^% 
by  65? 

81.  A  horse  and  carriage  are  worth  $  246.  If  -|  of  the 
value  of  the  horse  equals  |  of  the  value  of  the  carriage, 
what  is  the  value  of  each  ? 

82.  A  and  B  invest  $8800  in  business.  62^%  of  A's 
investment  equals  75%  of  B's.  How  much  does  each 
invest  ? 

83.  What  per  cent  of  the  circumference  of  a  circle 
2  inches  in  diameter  is  the  circumference  of  a  circle  3 
inches  in  diameter  ?     (See  problem  46,  page  33.) 

84.  What  per  cent  of  A's  money  is  B's,  if  A  has  $  50, 
and  B  has  10  fo  more  ?     What  per  cent  of  B's  is  A's  ? 

85.  What  per  cent  of  A's  money  is  B's  if  A  has  33 J  J6 
more  than  B  ?     What  per  cent  of  B's  is  A's  ? 


SECTION   VIII 

INTRODUCING  GENERAL  COMPOUND  NUMBERS  AND 
DEVELOPING  RELATIONS 

1.  What  is  the  ratio  of  8  in.  to  12  in.  ?  Of  2  ft.  to 
3  ft.  ?     Of  3  yd.  to  5£  yd.  ?     Of  32  rd.  to  320  rd.  ? 

2.  What  is  the  ratio  of  8  in.  to  1  ft.  ?  Of  2  ft.  to  1  yd.  ? 
Of  3  yd.  to  1  rd  ?     Of  32  rd.  to  1  mi.  ? 

3.  What  is  the  ratio  of  1  yd.  to  6  in.  ?  Of  1£  yd.  to 
6  in.  ?     Of  2  yd.  to  6  in.  ? 

4.  A  boy  measures  a  stick  with  a  6-inch  rule,  and  finds 
that  he  has  to  apply  the  rule  exactly  12  times.  How 
many  yards  long  is  the  stick  ? 

5.  How  many  steps  2  ft.  4  in.  long  must  a  man  take 
to  walk  a  distance  of  28  ft.  ?  What  is  the  ratio  of  28  ft. 
to  21  ft.  ? 

6.  How  many  inches  are  there  in  1  yd.  2  ft.  6  in.  ?  In 
1  yd.  1  ft.  3  in.  ?  In  1  rd.  ?  In  1  rd.  2  in.  ?  In  1  rd. 
1  yd.  1  ft.  1  in.  ? 

7.  Change  66  in.  to  integers  of  higher  denominations. 
(^66  in.  =  1  yd.  2  ft.  6  in.)  Change  200  in.  to  integers 
of  higher  denominations.     198  in.     51  in.     247  in. 

56 


GENERAL   COMPOUND  NUMBERS  57 

8.  How  many  feet  are  there  in  1  rd.  ?     In  2  rd.  ?     In 
4  rd.  ?     In  6  rd.  ?     In  1  rd,  3  yd.  ?     In  1  rd.  5  yd.  ?     In 

6  rd.  2  ft.  ? 

9.  Reduce  to  integers  of  higher  denominations:  33  ft.; 
99  ft.;  66  ft.;  25$  ft.;  100  ft.;  31 J  ft. 

10.  Five  planks,  5  ft.  6  in.,  7  ft.  3  in.,  6  ft.  3  in.,  5  ft. 

7  in.,  and  8  ft.  5  in.  long,  respectively,  are  laid  end  to  end. 
How  many  yards  along  the  ground  will  they  reach  ? 

11.  A  certain  fence  is  made  of  boards  8  in.  wide  stand- 
ing upright.  How  many  boards  are  there  in  6  yd.  2  ft.  of 
the  fence  ? 

12.  How  many  feet  will  a  boy  walk  in  taking  10  steps 
of  26  in.  each  ? 

13.  What  is  the  ratio  of  1  pt.  (dry  measure)  to  2  pt.? 
Of  3  qt.  to  8  qt.?     Of  1  pk.  to  4  pk.? 

14.  What  is  the  ratio  of  1  pt.  to  1  qt.?  Of  3  qt.  to 
1  pk.?     Of  1  pk.  to  1  bu.? 

15.  WThat  is  the  ratio  of  1  bu.  to  1  qt.?  Of  1  bu.  to 
1  pt.?  Of  1  bu.  to  2  pt.?  Of  1  bu.  to  2  qt.?  To  3  qt.? 
To  8  qt.?  To  1  pk.?  To  1  pk.  1  qt.?  To  2  pk.  3  qt.? 
To  1  pk.  1  qt.  1  pt.? 

16.  How  many  times  must  a  half -peck  measure  be  filled 
in  measuring  2^  bu.  of  beans  ? 

17.  Reduce  to  quarts :  4  pk.  2  qt. ;  1  bu.  1  pk. ;  2  bu. 
2pk.  2qt.;  1  bu.  3  pk.  5  qt. 

18.  Reduce  to  integers  of  higher  denominations :  48  qt. ; 
25  pt. ;  57  qt. ;  69  qt. 

19.  What  is  the  ratio  of  1  gi.  (liquid  measure)  to  4  gi.? 
Of  1  pt.  to  2  pt.?    Of  1  qt.  to  4  qt.?    Of  1  gal.  to  3£  gal.? 


58  GENERAL   COMPOUND  NUMBERS 

20.  What  is  the  ratio  of  1  gi.  to  1  pt.?  Of  1  pt.  to 
1  qt.?     Of  1  qt.  to  1  gal.?     Of  1  gal.  to  1  bbl.? 

21.  Give  the  ratio  of  1  gal.  to  1  qt.  To  1  gi.  To  1  pt. 
Of  lqt.  tolpt.     Tolgi. 

22.  Give  the  ratio  of  1  gal.  to  1  gal.  2  qt.  To  1  gal. 
1  qt.  1  pt.  To  2  gal.  2  gi.  Of  1  qt.  to  2  qt.  1  pt.  To 
1  gal.  1  qt.  1  pt.  To  1  pt.  3  gi.  Of  1  pt.  to  1  gal.  3  qt. 
To  2  qt.  1  pt.     To  1  pt.  2  gi. 

23.  How  many  times  can  a  dipper  holding  1  pt.  1  gi. 
be  filled  from  a  can  containing  3  gal.  1  pt.? 

24.  A  milkman  has  in  his  wagon  5  cans.  The  first  con- 
tains 10  gal.  of  milk;  the  second  4  gal.  2  qt.;  the  third 
1  gal.  2  qt. ;  the  fourth  4  gal.  1  pt.;  and  the  fifth  1  gal. 
3  qt.  1  pt.     How  many  gallons  of  milk  has  he  ? 

25.  How  many  bottles  holding  1  pt.  3  gi.  can  be  filled 
from  a  cask  containing  1  gal.  of  wine  ? 

26.  How  many  inch  squares  does  a  twelve-inch  square 
contain  ?  What  is  the  ratio  of  1  sq.  ft.  to  1  sq.  in.?  Of 
1  sq.  in.  to  1  sq.  ft.?  Of  36  sq.  in.  to  1  sq.  ft.?  Of  72 
sq.  in.  to  1  sq.  ft.?     Of  108  sq.  in.  to  1  sq.  ft.? 

27.  What  is  the  ratio  of  1  sq.  ft.  to  1  sq.  yd.?  Of  1 
sq.  yd.  to  1  sq.  ft.? 

28.  How  many  square  inches  are  there  in  a  2-inch 
square  ?  In  a  3-inch  square  ?  A  4-inch  square  ?  A 
5-inch  square?     A  10-inch  square? 

29.  What  is  the  ratio  of  a  2-inch  square  to  a  3-inch 
square  ?     To  a  4-inch  square  ?     To  a  6-inch  square  ? 


GENERAL   COMPOUND  NUMBERS  59 

30.  Give  the  ratio  of :  A  3-inch  square  to  a  2-inch 
square  ;  a  4-inch  square  to  a  2-inch  square  ;  a  3-inch 
square  to  a  5-inch  square  ;  a  5-inch  square  to  a  10-inch 
square  ;  a  12-inch  square  to  a  36-inch  square. 

31.  Rectangle  a  is  3  in.  long  and  2  in.  wide;  rectangle  b 
is  5  in.  long  and  3  in.  wide.  What  is  the  ratio  of  a  to  b  ? 
Oibtoa  ?     Of  1  of  b  to  -i  of  a  ?     Of  1  of  a  to  1  of  b  ? 

32.  At  the  rate  of  10  shingles  to  the  square  foot,  how 
many  shingles  are  necessary  for  a  roof  20  ft.  wide  and 
50  ft.  long  ? 

33.  A  cube  is  a  solid  having  six  equal  square  faces.  A 
1-inch  cube  (or  cubic  inch)  has  each  edge  1  inch  in  length. 
How  many  1-inch  cubes  are  there  in  a  2-inch  cube  ?  How 
many  cubic  inches  are  there  in  a  2-inch  cube  ?  In  a 
3-inch  cube  ?  A  4-inch  cube  ?  A  5-inch  cube  ?  A  10- 
inch  cube  ?     A  12-inch  cube  ? 

34.  What  is  the  ratio  of :  A  2-inch  cube  to  a  3-inch 
cube  ?  To  a  4-inch  cube  ?  To  a  5-inch  cube  ?  To  a 
10-inch  cube  ? 

35.  What  is  the  ratio  of  a  3-inch  cube  to  a  4-inch  cube? 
To  a  5-inch  cube  ?  To  a  6-inch  cube  ?  To  a  10-inch  cube  ? 
Of  a  4-inch  cube  to  a  5-inch  cube  ?    To  a  10-inch  cube  ? 

36.  How  many  cubic  inches  are  there  in  a  block  of 
wood  4  in.  long,  3  in.  wide,  and  1  in.  thick  ?  In  a  block 
4  in.  long,  3  in.  wide,  and  2  in.  thick  ?  In  a  block  4  in. 
long,  3  in.  wide,  and  5  in.  thick  ? 

37.  How  many  cubic  inches  are  there  in  a  cubic  foot  ? 
How  many  cubic  feet  are  there  in  a  3-foot  cube  ?  In  a 
cubic  yard  ? 


60  GENERAL   COMPOUND  NUMBERS 

38.  In  digging  a  ditch  20  ft.  long,  2  ft.  wide,  and  1  ft. 
deep,  how  many  cubic  feet  of  dirt  are  handled?  How 
many,  if  the  ditch  is  2  ft.  deep?  3  ft.  deep?  4  ft.? 
5  ft.? 

39.  A  wheat  bin  is  10  ft.  long,  8  ft.  wide,  and  5  ft.  deep. 
How  many  cubic  feet  of  wheat  will  it  hold  ? 

40.  Allowing  1J  cu.  ft.  of  wheat  to  the  bushel,  how  many 
bushels  will  the  bin  hold  ? 

41.  How  many  bushels  of  oats  can  be  put  into  a  box 
4  ft.  long,  2  ft.  wide,  and  18  in.  deep  ? 

42.  There  are  about  7J  gal.  in  a  cubic  foot.  How  many 
gallons  will  a  tank  2  ft.  x  3  ft.  x  4  ft.  hold  ? 

43.  An  aquarium  measures  4  ft.  long,  2J  ft.  wide,  and 
18  in.  deep.  How  many  gallons  of  water  are  necessary  to 
fill  it  half  full  ? 

44.  A  gallon  contains  231  cubic  inches.  How  deep  must 
a  box  be  to  hold  exactly  a  gallon,  if  it  is  11  in.  long  and 
7  in.  wide  ? 

45.  What  is  the  ratio  of  3  bu.  of  wheat  to  3|  cu.  ft.  of 
wheat  ? 

46.  Give  the  ratio  of  a  5-inch  square  to  a  6-inch  square. 
Of  a  4-inch  square  to  a  5-inch  square.  Of  a  4-inch  cube 
to  a  5-inch  cube.     Of  a  10-inch  cube  to  a  5-inch  cube. 

47.  How  many  cubic  inches  are  there  in  a  block  of 
wood  12  in.  long,  5J  in.  wide,  and  1 J  in.  thick  ? 

48.  What  is  the  cost  of  digging  a  cellar  10  ft.  deep, 
10  ft.  wide,  and  27  ft.  long  at  $  3.50  per  cubic  yard  ? 

49.  How  many  2-bushel  grain  sacks  can  be  filled  from 
a  bin  3  ft.  x  4  ft.  x  5  ft.  full  of  wheat  ? 


GENERAL   COMPOUND  NUMBERS  61 

50.  About  how  many  gallons  of  milk  will  a  cheese  vat 
10  ft.  x  4  ft.  x  3  ft.  hold  ? 

51.  If  the  bottom  of  a  jar  has  an  area  of  21  sq.  in., 
how  deep  must  the  jar  be  to  hold  exactly  a  gallon  ? 

52.  A  steel  plate  20  in.  long  and  1  ft.  wide  weighs  8 
oz.  to  the  square  inch  of  surface.  How  many  pounds 
does  the  plate  weigh  ? 

53.  How  many  pieces  1  ft.  square  and  1  in.  thick  are 
there  in  an  inch  board  1  ft.  wide  and  10  ft.  long  ?  In  a 
2-inch  plank  of  the  same  width  and  length  ? 

54.  How  many  units  1  in.  thick,  having  a  surface  of 
1  sq.  ft.,  are  there  in  a  board  1  in.  thick,  6  in.  wide,  and 
12  ft.  long  ? 

Such  units  are  called  board  feet.  The  board  foot  is  used  in  meas- 
uring lumber. 

55.  How  many  board  feet  are  there  in  a  plank  8  in. 
wide,  15  ft.  long,  and  2  in.  thick  ? 

The  surface  of  the  plank  =  10  sq.  ft.  Since  it  is  2  in.  thick,  for 
each  square  foot  there  are  2  bd.  ft.  Therefore  there  are  20  bd.  ft.  in 
the  plank. 

56.  How  many  board  feet  are  there  in  a  2-inch  plank 
6  in.  wide  .and  16  ft.  long  ?     In  10  such  planks  ? 

57.  How  many  board  feet  are  there  in  seven  2  by  4's 
(i.e.  plank  2  in.  thick  and  4  in.  wide)  12  ft.  long?  In 
twenty  2  by  4's  12  ft.  long  ?     In  ten  2  by  6's  20  ft.  long  ? 

58.  How  many  feet  (board  feet)  are  there  in  ten  8  by 
8's  18  ft.  long  ?     In  ten  4  by  4's  ? 

59.  How  many  feet  of  lumber  are  necessary  to  plank 
the  floor  of  a  stall  8  ft.  x  10  ft.  with  2  by  6's  ?  How 
much  will  it  cost  at  $  20  per  M.  ? 


62  GENERAL   COMPOUND  NUMBERS 

60.  Boards  less  than  an  inch  thick  are  reckoned  as 
though  they  were  an  inch  thick.  What  is  the  value  of 
ten  |-inch  black  walnut  boards,  10  in.  wide  and  12  ft. 
long,  at  $  100  per  M.  ? 

61.  At  the  rate  of  10  shingles  to  the  square  foot,  how 
much  will  shingles  cost  for  a  roof  20  ft.  by  50  ft.  at  $  3 
per  M.  ? 

62.  At  $2.50  per  M.,  how  much  will  the  shingles  cost 
for  a  roof  of  two  parts,  each  40  ft.  by  20  ft.  ? 

63.  How  much  would  it  cost  to  cover  the  same  roof 
with  inch  boards  at  $  10  per  M.  ? 

64.  What  is  the  ratio  of  2  bu.  3  pk.  to  1  pk.  3  qt.  ? 
Of  1  bu.  1  pk.  1  qt.  to  2  bu.  2  pk.  2  qt.  ?  Of  4  gal.  1  pt. 
to  3  qt.  ?     Of  4  rd.  2J  ft.  to  8  rd.  1  yd.  2  ft.  ? 

65.  Give  the  ratio  of : 

2  lb.  Avoir,  to  4J  lb.  Avoir. 

6  oz.  Avoir,  to  2  lb.  Avoir. 

1  lb.  4  oz.  Avoir  to  3  lb.  Avoir. 

2  lb.  12  oz.  Avoir,  to  3  lb.  8  oz.  Avoir. 

7  oz.  Avoir,  to  2  lb.  3  oz.  Avoir. 
2  lb.  Troy  to  4J  lb.  Troy. 

6  oz.  Troy  to  2  lb.  Troy. 

1  lb.  4  oz.  Troy  to  3  lb.  Troy. 

2  lb.  6  oz.  Troy  to  3  lb.  4  oz.  Troy. 

7  oz.  Troy  to  2  lb.  4  oz.  Troy. 

Suggestion.  If  the  teacher  has  time,  Liquid  Measure  and  the 
Metric  System  can  be  easily  treated  in  a  manner  similar  to  that  in  65 
above. 


GENERAL   COMPOUND  NUMBERS  63 

€6.  How  many  bushels  of  wheat  are  there  in  a  bin 
5 J  ft.  x  6  ft.,  and  5  ft.  deep,  when  it  is  f  full?  (See 
problem  40.) 

67.  What  is  the  cost  of  digging  a  cellar  6  ft.  deep,  10 
ft.  long,  and  9  ft.  wide,  at  $2.70  per  cu.  yd.  ? 

68.  About  how  many  gallons  of  water  will  a  tank  hold, 
whose  inside  measurements  are  4  ft.  x  3|  ft.  x  1J  ft.?  (See 
problem  42.) 

69.  At  $10  per  M.,  what  is  the  cost  of  eight  2xl0's 
12  ft.  long,  and  ten  8  x  8's  18  ft.  long  ? 

70.  At  the  rate  of  1000  shingles  to  the  100  sq.  ft.,  how 
many  shingles  are  necessary  for  a  roof  40  ft.  x  80  ft.  ? 

71.  The  area  of  the  bottom  of  a  jar  is  77  sq.  in.  How 
deep  must  it  be  to  hold  exactly  2  gal.  ? 

72.  How  many  times  can  a  cask  holding  3|  gal.  be  filled 
from  a  tank  4  ft.  long,  3  ft.  wide,  and  2  ft.  deep  ? 

73.  How  many  cubic  feet  are  there  in  a  block  of  stone 
30  in.  x  24  in.  x  18  in.? 

74.  At  40  lb.  to  the  cubic  foot,  what  is  the  weight  of  a 
block  of  wood  6  ft.  long,  18  in.  wide,  and  1J  ft.  thick  ? 

75.  How  many  feet  of  lumber  are  necessary  to  cover 
with  2-in.  plank  a  surface  20  ft.  x  80  ft.  ?  How  much  will 
the  lumber  cost  at  $20  per  M.? 

76.  How  many  revolutions  does  a  wagon  wheel  3J  ft. 
in  diameter  make  in  going  12  rd.  ?  (See  problem  46, 
page  33.) 

77.  At  $  3  per  cubic  yard,  how  much  will  it  cost  to  dig 
a  cellar  30  ft.  x  33  ft.,  and  12  ft.  deep  ? 


64  GENERAL   COMPOUND  NUMBERS 

78.  How  many  bushels  of  grain  can  be  put  into  a  bin 
10  ft.  x  8  ft.,  and  6  ft.  deep  ?  How  many  bushels  can  be 
put  into  a  bin  20  ft.  x  16  ft.  x  12  ft.  ? 

79.  If  wall  paper  is  18  in.  wide,  how  many  strips  are 
needed  to  cover  a  wall  30  ft.  long  ?  How  many  strips  are 
needed  for  the  four  walls  of  a  room  30  ft.  long  and  24  ft. 
wide,  making  no  allowance  for  doors  and  windows  ? 

80.  How  many  more  steps  must  be  taken  in  walking  a 
mile  if  the  steps  are  2  ft.  long,  than  if  they  are  3  ft.  long  ? 

81.  At  the  rate  of  8  mi.  per  hour,  how  far  will  a  horse 
travel  in  4  hr.  35  min.  ? 

82.  At  the  rate  of  37|  mi.  in  90  min.,  how  far  will  a 
train  run  in  3^  hr.  ? 

83.  Reduce  208  in.  to  integers  of  higher  denomination. 
92  ft.;  125  in.;  144  in. 

84.  At  $10  per  M.,  find  the  cost  of  ten  4  x  4's  18  ft. 
long,  and  thirty  2  x  6's  12  ft.  long. 


SECTION   IX 


INTRODUCING  THE  STUDY  OF  SURFACES  AND 
DEVELOPING  RELATIONS 


1.  If  the  above  rectangle  is  8  units  long,  how  many 
square  units  are  there  along  one  side  ?  If  it  is  4  units 
wide,  how  many  square  units  are  there  along  one  end  ? 
How  many  rows  are  there .  of  8  square  units  each  ?  How 
many  square  units  are  there  in  the  whole  rectangle  ?  How 
many  rows  are  there  of  4  square  units  each  ?  How  many 
square  units  are  there  in  the  rectangle  ? 

2.  Does  4x8  equal  the  number  of  square  units  in  a 
rectangle  8  units  long  and  4  units  wide  ?  How  many 
square  units  are  there  in  a  rectangle  10  units  long  and  7 
units  wide  ? 

3.  The  number  of  square  units  in  the  area  of  a  rectangle 
is  equal  to  the  product  of  the  numbers  of  the  like  units  in 
the  length  and  width.  What  is  the  area  of  a  rectangle 
6  in.  long  and  4  in.  wide  ?     1  ft.  long  and  5  in.  wide  ? 

MON.  MENT.   AR. 5  65 


66 


THE  STUDY  OF  SURFACES 


4.  A  parallelogram  is  a  four-sided  figure  whose  opposite 
sides  are  equal  and  parallel.  Is  a  rectangle  a  parallelo- 
gram ?  The  base  is  the  side  on  which  the  parallelogram 
seems  to  stand.  The  altitude  is  the  perpendicular  dis- 
tance from  the  base  to  the  side  opposite.  In  figure  a 
what  is  the  base  ?  The  altitude  ?  In  figure  b  what  is  the 
base  ?     What  two  lines  represent  the  altitude  ? 

5.  If  in  figure  b  we  cut  off  a  three-sided  portion  at  the 
right,  and  add  on  an  equal  three-sided  portion  at  the  left, 
what  kind  of  a  figure  have  we  ?  Compare  the  area  of  the 
rectangle  with  the  area  of  parallelogram  b. 

6.  If  the  rectangle  is  8  units  long  and  4  units  wide, 
how  many  square  units  are  there  in  its  area?  If  the 
parallelogram  has  a  base  of  8  units,  and  an  altitude  of 
4  units,  how  many  square  units  are  there  in  its  area  ? 

7.  The  area  of  a  parallelogram  is  represented  by  the 
product  of  the  numbers  of  like  units  in  the  base  and  in  the 
altitude.  What  is  the  area  of  a  parallelogram  whose  base 
is  7  in.  and  altitude  4  in.  ?  Base  10  in.  and  altitude  6  in.  ? 
Base  20  in.  and  altitude  7  in.  ? 


B  b  '       b 

8.    A  triangle  is  a  three-sided  figure.     Its  base  is  usu- 
ally considered  to  be  the  side  it  seems  to  stand  on,  although 


THE  STUDY  OF  SURFACES  '  67 

any  side  may  be  considered  the  base.  Its  altitude,  AB, 
is  its  height,  measured  from  the  highest  point  to  the  base. 
If  a  line  is  drawn  from  corner  to  corner  of  a  parallelo- 
gram, into  what  two  figures  is  the  parallelogram  divided  ? 
Compare  triangle  a  with  triangle  b.  Compare  a  with  the 
parallelogram.     Compare  b  with  the  parallelogram. 

9.  Is  the  altitude  of  triangles  a  and  b  the  same  as  that 
of  the  parallelogram  ?  Are  the  bases-  the  same  ?  If  the 
base  of  the  parallelogram  is  7  and  its  altitude  3,  what  is  its 
area?    What,  then,  is  the  area  of  triangle  a ?   Of  triangle  b ? 

10.  The  area  of  a  triangle  is  represented  by  one  half  the 
product  of  the  numbers  of  like  units  in  the  base  and  in  the 
altitude.  The  base  of  a  certain  triangle  is  7  units,  its 
altitude  3  units.     What  is  its  area  ? 

11.  Give  the  area  of  a  triangle  whose  base  is  7  in.  and 
altitude  3  in.  Base  10  in.  and  altitude  4  in.  Base  6  ft. 
and  altitude  3  ft.     Base  4  rd.  and  altitude  3  rd. 

12.  What  is  the  area  of  a  rectangle  12  ft.  long  and  7  ft. 
wide?  Of  a  parallelogram  whose  base  is  12  ft.  and  alti- 
tude 7  ft.  ?  Of  a  triangle  whose  base  is  12  ft.  and  alti- 
tude 7  ft.  ?  What  is  the  ratio  of  the  rectangle  to  the 
parallelogram?  Of  the  parallelogram  to  the  triangle? 
Of  the  rectangle  to  the  triangle? 

13.  A  rectangular  piece  of  land  is  divided  into  two 
parts  by  a  diagonal  line  (one  running  from  corner  to 
corner).  If  the  base  of  the  rectangle  is  40  rd.  and  its 
altitude  20  rd.,  what  is  the  area  of  each  of  the  parts? 

14.  A  trapezoid  is  a  four-sided  figure  having  two  and 
only  two  sides  parallel.     These  sides  are  called  the  upper 


68 


THE  STUDY  OF  SURFACES 


base  and  the  lower  base.  The  altitude  is  the  perpen- 
dicular distance  between  the  parallel  sides.  If  a  diagonal 
is  drawn,  into  what  does  it  divide  the  trapezoid?  If  the 
upper  base  is  4  and  the  altitude  is  2,  what  is  the  area  of 


triangle  a?  If  the  lower  base  is  3,  what  is  the  area  of 
triangle  b?  Of  both  triangles?  The  area  of  the  trape- 
zoid? Will  you  get  the  same  area  of  the  trapezoid  if  you 
take  half  the  sum  of  the  two  bases  times  the  altitude  ? 

15.  The  area  of  a  trapezoid  is  represented  by  the  product 
of  the  number  of  like  units  in  half  the  sum  of  the  two  bases 
and  in  the  altitude.  What  is  the  area  of  a  trapezoid 
whose  bases  are  4  units  and  3  units,  and  altitude  2  units  ? 
Of  one  whose  bases  are  4  in.  and  3  in.,  and  altitude 
2  in.? 

16.  Find  the  area  of  a  trapezoid  having : 

Bases  6  in.  and  8  in.,  and  altitude  4  in. 
Bases  4  ft.  and  5  ft.,  and  altitude  3  ft. 
Bases  2  ft.  and  18  in.,  and  altitude  15  in. 

17.  A  certain  rectangle  has  an  area  of  72  sq.  in.  If  its 
length  is  1  ft.,  how  wide  is  it? 

18.  A  parallelogram  having  an  altitude  of  7  ft.  has  an 
area  of  81  sq.  ft.     How  long  is  the  base  ? 


THE  STUDY  OF  SURFACES 


69 


19.  The  area  of  a  certain  triangle  is  40  sq.  in.  If  the 
base  is  10  in.,  what  is  the  altitude? 

20.  The  area  of  a  certain  trapezoid  is  85  sq.  ft.  If  one 
base  is  20  ft.,  and  the  other  14  ft.,  what  is  the  altitude  of 
the  trapezoid  ? 

21.  Think  of  a  circle  divided  into  a  number  of  equal 
triangles  as  in  Fig.  1.  Is  the  area  of  all  the  triangles 
greater  or  less  than  the  area  of  the  circle?  Are  the  alti- 
tudes of  the  triangles  equal?     Is  the  altitude  a  greater 


Fig.  i. 


Fig.  2. 


or  less  than  the  radius  of  the  circle  ?  Is  the  sum  of  the 
bases  (the  sides  next  to  the  circumference)  greater  or  less 
than  the  circumference?  If  the  number  of  triangles  be 
doubled,  is  their  area  nearer  to  that  of  the  circle  ?  Is  the 
altitude  a  more  nearly  equal  to  the  radius?  Is  the  sum 
of  the  bases  more  nearly  equal  to  the  circumference  ?  If 
you  think  of  the  number  of  triangles  indefinitely  increased, 
what  will  be  true  of  their  area,  altitude,  and  the  sum  of 
their  bases,  as  compared  with  the  area,  radius,  and  circum- 
ference of  the  circle  respectively?  Can  they  be  made  as 
nearly  equal  as  we  please? 

22.    Think  of  the  circle  in  Fig.  2  above  as  unrolled. 
The  sum  of  the  bases  of  the  triangles  may  be  represented 


70 


THE  STUDY  OF  SURFACES 


as  below  by  a  straight  line  (nearly)  equal  to  the  circum- 
ference. How  is  the  area  of  each  triangle  found  ?  Is  the 
area  of  all  of  them  equal  to  ^  the  product  of  the  sum  of 
the  bases  by  the  altitude?     Since  this  is  true,  and  since 


Circumference 


i  Circumference 


the  sum  of  the  bases  is  nearly  equal  to  the  circumference, 
and  the  altitude  to  the  radius,  is  the  area  of  the  circle 
nearly  equal  to  ^  the  product  of  the  circumference  by 
the  radius,  or  the  product  of  \  the  circumference  by  the 
radius?  By  increasing  the  number  of  triangles,  is  the  re- 
sult more  nearly  correct?  (Geometry  shows  that  the  area 
of  the  circle  is  exactly  equal  to  J  the  product  of  the  circum- 
ference by  the  radius.^  In  the  above  figure  is  represented 
a  parallelogram  nearly  equal  in  area  to  the  circle.  How 
can  you  find  its  area  ? 

23.  Find  the  area  of  a  circle  whose  circumference  is  44  in. 
and  radius  7  in.  Of  one  whose  circumference  is  22  in.  and 
radius  3^  in.  State  the  dimensions  of  the  rectangle  equal 
in  area  to  each  circle. 


A  rectangle  7  in.  x  22  in.  is  equal  to  the  first  circle;  that  is,  one 
whose  width  is  the  radius,  and  whose  length  is  \  the  circumference. 


THE  STUDY  OF  SURFACES 


71 


24.  What  is  the  ratio  of  the  circumference  to  the  diame- 
ter? (Nearly  -2Y2-.)  How  many  times  the  radius  is  the 
diameter?  How  many  times  the  radius  is  the  circumfer- 
ence ?  What  is  the  circumference  of  a  circle  whose  radius 
is  7  in.  ?  Radius  8J  in.  ?  Radius  14  ft.?  Radius  17£  ft.  ? 
Radius  21  ft.  ?  How  long  must  a  rectangle  7  in.  wide 
be  to  be  equal  to  the  first  circle  ?  Make  a  similar  state- 
ment regarding  each  of  the  other  circles. 


it  x  r 


\  Circumference  =  n  x  r 


25.  Find  the  area  of  a  circle  whose  radius  is  7  in. 
What  rectangle  is  it  equal  to  ?  Into  how  many  squares 
7  in.  on  each  side  can  this  rectangle  be  divided  ?  (Each 
square  may  be  thought  of  as  the  square  of  the  radius  rep- 
resented above  by  r2.)  Is  the  area  of  the  rectangle  equal 
to  3^  x  the  square  of  7  in.  ?  Using  the  symbol  ir  (read 
pi)  for  34-,  is  the  area  of  the  rectangle  equal  to  nr  x  49  sq. 
in.,  or  it  x  r2,  or  77T2?  Is  the  area  of  the  circle  also  equal 
to  irr2  ?  (This  way  of  finding  the  area  is  easier  than  find- 
ing ^  the  product  of  the  circumference  by  the  radius,  since 
the  radius,  not  the  circumference,  is  usually  given.) 

26.  Find  the  area  of  a  circle  whose  radius  is  10^  in. 

11     3 

(  Area  =  7rr2,  or,  ^  x  ^  x  —  =  346£,  number  of  sq.  in.  J 


72 


THE  STUDY  OF  SURFACES 


Of  one  whose  radius  is  3^  in 
rd. 

yd 


;   14  in.;   7ft.;   31  yd.;  14' 
Of  one  whose  diameter  is  7  in. ;  28  in. ;   14  ft.  ;   7 
28  rd. :  7  ft. 


27.  What  is  the  ratio  of  the  length  of  rectangle  a  to  the 
length  of  rectangle  b  ?  Of  the  width  of  rectangle  a  to  the 
width  of  rectangle  b  ?  Of  the  length  x  the  width  of  rec- 
tangle a  to  the  length  x  the  width  of  rectangle  b  ?  Of 
rectangle  a  to  rectangle  b  ?     Of  b  to  a  ? 

28.  If  rectangle  a  is  36  in.  long  and  25  in.  wide,  and 
rectangle  b  is  27  in.  long  and  20  in.  wide,  what  is  the 
ratio  of  the  length  of  a  to  the  length  of  b  ?  Of  the  width 
of  a  to  the  width  of  b  ?  Of  the  area  of  a  to  the  area  of  b  ? 
Of  b  to  a  ? 

29.  Give  the  ratio  of : 

A  rectangle  12  in.  by  9  in.  to  a  rectangle  8  in.  by  6  in. 
A  rectangle  5  in.  by  7  in.  to  a  rectangle  7 J  in.  by  10 J  in. 
A  rectangle  18|  in.  by  25  in.  to  a  rectangle  25  in.  by  16|  in. 
A  rectangle  12J  ft.  by  8^  ft.  to  a  rectangle  37^  ft.  by  16|  ft. 
A  rectangle  31^  rd.  by  20  rd.  to  a  rectangle  37J  rd.  by  30  rd. 


THE  STUDY  OF  SURFACES  73 

A  rectangle  83^  yd.  by  33^  yd.  to  a  rectangle  33J  yd.  by 

8*  yd. 

A  rectangle  56^  ft.  by  41|  ft.  to  a  rectangle  62^  ft.  by  58^  ft. 

30.  How  many  rectangles  8  in.  x  5  in.,  will  a  rectangle 
32  in.  x  15  in.  contain  ? 

31.  What  is  the  area  of : 

A  parallelogram  whose  base  is  8  ft.  and  altitude  7  ft.  ? 

A  triangle  whose  base  is  8  ft.  and  altitude  7  ft.  ? 

» 

A  trapezoid  whose  bases  are  6  ft.  and  4  ft.,  and  altitude 
10  ft.  ? 

A  triangle  whose  base  is  3  ft.  and  altitude  16  in.  ? 

A  rectangle  4  ft.  by  4  yd.  ? 

A  circle  whose  radius  is  7  in.  ? 

A  circle  whose  diameter  is  7  in.  ? 

A  trapezoid  whose  bases  are  8  ft.  and  8  yd.,  and  altitude 
10  ft.  ? 


n 


32.  What  is  the  ratio  of  the  base  of  a  to  the  base  of  b  ? 
Of  the  altitude  of  a  to  the  altitude  of  b  ?  If  the  ratio  of 
the  bases  is  f ,  and  the  ratio  of  the  altitudes  is  2,  what  is 
the  ratio  of  the  base  x  altitude  of  a,  to  the  base  x  altitude 
of  b  ?  Or,  what  is  the  ratio  of  the  area  of  a  to  the  area  of 
b  ?     Of  b  to  a  ?  (f  x  i  -  ?) 


74 


THE  STUDY  OF  SUBFACES 


33.    Give  the  ratio  of  the  area  of  parallelogram  a  to 
parallelogram  b  in  the  following.     Also  of  b  to  a. 


Base  .8  in.,  alt.  6  in. 
Base  12  in.,  alt.  8  in. 
Base  18f  ft.,  alt.  16$  ft. 
Base  911  rd.,  alt.  56{  rd. 
Base  66%  yd.,  alt.  25  yd. 
Base  50  ft.,  alt.  62£  ft. 
Base  100  ft.,  alt.  300  ft. 
Base  41$  in.,  alt.  50  in. 


Base  4  in.,  alt.  3  in. 
Base  9  in.,  alt.  5  in. 
Base  25  ft.,  alt.  33£  ft. 
Base  75  rd.,  alt.  31Jrd. 
Base  100  yd.,  alt.  37£  yd. 
Base  66%  ft.,  alt.  37J  ft. 
Base  200  ft.,  alt.  150  ft. 
Base  33£  in.,  alt.  125  in. 


34.  What  is  the  ratio  of  the  area  of  parallelogram  a  to 
the  area  of  parallelogram  b  ?  If  a  is  3  x  6,  what  is  the 
ratio  of  J  of  a  to  ^  of  b  ?  Triangle  a  is  what  part  of 
parallelogram  a?  Triangle  b  is  what  part  of  parallelo- 
gram b  ?  What,  then,  is  the  ratio  of  triangle  a  to 
triangle  b  ? 

35.  What  is  the  ratio  of  a  triangle  whose  base  is  6, 
and  altitude  3,  to  one  whose  base  is  4  and  altitude  1J? 

ct  * !  =  ?)• 

Why  is  it  not  necessary  to  consider  the  \  in  comparing  two 
triangles  ?     (See  problem  10.) 


THE  STUDY  OF  SUBFACES  75 

36.    What  is  the  ratio  of  triangle  a  to  triangle  b  in  the 
following  ?     Of  b  to  a  ? 


a 


b 


Base  8  in.,  alt.  6  in.  Base  4  in.,  alt.  3  in. 

Base  12  in.,  alt.  8  in.  Base  9  in.,  alt.  5  in. 

Base  18f  ft.,  alt.  16$  ft.  Base  25  ft.,  alt.'  83}  ft. 

Base  91  J-  rd.,  alt.  56^  rd.  Base  75  rd.,  alt.  81}  rd. 

Base  34  in.,  alt.  2^  in.  Base  5  in.,  alt.  5  in. 

Base  18f  in.,  alt.  25  in.  Base  25  in.,  alt.  16$  in. 

Base  121  ft.,  alt.  8}  ft.  Base  37*  ft.,  alt.  16$  ft. 

Base  561  ft.,  alt.  41$  ft.  Base  621  ft.,  alt.  581  ft. 

37.  Find  the  ratio  of  a  7-in.  circle  to  a  14-in.  circle. 

Find  the  dimensions  of  the  rectangles  equal  to  the  two  circles. 
(See  problems  24  and  25.)  If  the  altitudes  are  7  in.  and  14  in.,  and 
the  bases  are  3f  x  7  in.,  and  3f  x  14  in.,  what  is  the  ratio  of  the  alti- 
tudes and  of  the  bases  ?  The  ratio  of  the  rectangles  ?  Of  the  circles  ? 
Since  the  ratio  of  the  altitudes  is  equal  to  the  ratio  of  the  bases,  can 
the  ratio  of  the  areas  be  found  by  squaring  the  ratio  of  the  altitudes, 
or  their  equal,  the  ratio  of  the  two  radii  ? 

38.  What  is  the  ratio  of  a  circle  of  radius  3^  ft.  to  one 
of  radius  7.  ft.  ?  Of  one  of  radius  6  ft.  to  one  of  radius  12 
ft.  ?     Of  one  of  radius  12J  ft.  to  one  of  radius  18|  ft.  ? 

What  is  the  ratio  of  the  radii?  If  the  ratio  of  the  radii  is  §,  what 
is  the  ratio  of  the  circles  ? 

39.  If  the  radius  of  one  circle  is.  twice  as  long  as  the 
radius  of  another,  what  is  the  ratio  of  the  area  of  the  first 
to  that  of  the  second  ?     Of  the  second  to  the  first  ? 


76  THE  STUDY  OF  SURFACES 

40.  If  the  area  of  one  circle  is  4  times  that  of  another, 
what  is  the  ratio  of  the  radius  of  the  first  to  that  of  the 
second  ?  Of  the  diameter  of  the  first  to  the  diameter 
of  the  second  ?  Of  the  circumference  of  the  first  to  the 
circumference  of  the  second  ? 

41.  What  is  the  ratio  of  the  area  of  a  circle  of  radius  1  ft. 
to  the  area  of  one  of  radius  2  ft.  ?  Of  one  of  radius  2  ft. 
to  one  of  radius  4  ft.  ?  Of  one  of  radius  3  ft.  to  one  of 
radius  6  ft.  ?  Of  one  of  diameter  1  ft.  to  one  of  diameter 
2  ft.  ?  Of  one  of  diameter  2  ft.  to  one  of  diameter  4  ft.  ? 
Of  one  of  diameter  3  ft.  to  one  of  diameter  6  ft.  ?  Of  one 
of  circumference  1  ft.  to  one  of  circumference  2  ft.  ?  Of 
one  of  circumference  2  ft.  to  one  of  circumference  4  ft.  ? 
Of  one  of  circumference  3  ft.  to  one  of  circumference 
6  ft.  ? 

42.  A  pipe  1  inch  in  diameter  carries  3  gal.  of  water  per 
minute.  At  that  rate  how  much  per  minute  will  a  2-inch 
pipe  carry  ?  A  4-inch  pipe  ?  An  8-inch  pipe  ?  A  ^-inch 
pipe  ?     A  ^-inch  pipe  ? 

43.  A  building  is  supplied  with  water  by  a  2-inch  pipe. 
How  many  J-inch  pipes  would  it  take  to  carry  the  same 
amount  of  water  running  at  the  same  rate  ? 

44."  What  is  the  ratio  of  the  area  of  a  circle  of  1  in.  ra- 
dius to  that  of  a  circle  of  3  in.  radius  ?  (See  problem  27.) 
Of  a  circle  of  2  in.  radius  to  one  of  6  in.  radius  ?  Of  a 
circle  of  3  in.  radius  to  one  of  9  in.  radius  ?  Of  a  circle 
of  any  radius  to  a  circle  of  3  times  that  radius  ?  Of  a 
circle  of  any  diameter  to  a  circle  of  3  times  that  diameter  ? 
Of  a  circle  of  any  circumference  to  a  circle  of  3  times 
that  circumference  ? 


THE  STUDY  OF  SURFACES  77 

45.  What  is  the  ratio  of  the  area  of  a  circle  of  radius  1 
in.  to  that  of  a  circle  of  radius  4  in.  ?  Of  a  circle  of  radius 
2  in.,  to  one  of  radius  8  in.  ?  Of  a  circle  of  radius  3  in., 
to  one  of  radius  12  in.  ?  Of  a  circle  of  any  radius,  to 
One  of  4  times  that  radius  ?  Of  a  circle  of  any  diameter, 
to  one  of  4  times  that  diameter  ?  Of  a  circle  of  any  cir- 
cumference, to  one  of  4  times  that  circumference  ? 

46.  The  square  of  a  number  is  the  product  obtained  by 
multiplying  the  number  by  itself.  What  is  the  square  of 
6  ?  Of  5  ?  Of  8  ?  Of  |  ?  Off?  The  square  of  6  may 
be  represented  thus  :    62  ;   of  -|,   (|)2,  etc.     72  =  ?    92  =  ? 

(t)2=?  (t)2=?  d)2=? 

47.  The  ratio  of  the  area  of  one  circle  to  that  of  another 
is  the  square  of  the  ratio  of  the  radius  of  the  first  to  the 
radius  of  the  second,  or  the  square  of  the  ratio  of  the 
diameter  of  the  first  to  the  diameter  of  the  second,  or 
the  square  of  the  ratio  of  the  circumference  of  the  first 
to  the  circumference  of  the  second.  Or,  The  ratio  of  one 
circle  to  another  equals  the  square  of  the  ratio  of  two  homol- 
ogous (like)  lines. 

Give  the  ratio  of : 

A  circle  of  radius  1  ft.  to  one  of  radius  2  ft. 

A  circle  of  diameter  1  ft.  to  one  of  diameter  2  ft. 

A  circle  of  circumference  1  ft.  to  one  of  circumference 
2  ft. 

A  circle  of  radius  2  ft.  to  one  of  radius  8  ft. 

The  ratio  of  the  radii  is  \.  Then  the  ratio  of  the  circles  is 
(i)2>  or  rV 


78  THE  STUDY  OF  SUBFACES 

A  circle  of  radius  2  ft.  to  one  of  radius  10  ft. 

A  circle  of  radius  2  ft.  to  one  of  radius  12  ft. 

A  circle  of  diameter  5  ft.  to  one  of  diameter  30  ft. 

The  ratio  of  the  diameters  is  }.     Then  the  ratio  of  the  circles  is 
G)2,  or  *V 

A  circle  of  circumference  »4  ft.  to  one  of  circumference 
10  ft. 
The  ratio  of  the  circumferences  is  f.     Then  the  ratio  of  the  circles 
is  (f  )2,  or  ^. 

A  circle  of  circumference  6^  ft.  to  one  of  circumference 

18|  ft. 
A  circle  of  diameter  2^  ft.  to  one  of  diameter  12J  ft. 
A  circle  of  radius  3^  ft.  to  one  of  radius  20  ft. 
A  circle  of  radius  2^  ft.  to  one  of  radius  18  ft. 
A  circle  of  circumference  2\  ft.  to  one  of  circumference 

18  ft. 
A  circle  of  radius  3  ft.  to  one  of  diameter  3  ft. 
A  circle  of  radius  3J  ft.  to  one  of  circumference  22  ft. 
A  circle  of  radius  100  ft.  to  one  of  diameter  200  ft. 


SECTION   X 

INTRODUCING  THE  STUDY  OF  SOLIDS   AND  DEVELOPING 

RELATIONS 


1.  Think  of  a  square  block  of  wood,  6  in.  wide,  6  in. 
long,  and  6  in.  high.  What  is  a  solid  of  this  shape  called  ? 
How  many  surfaces  has  it  ?  What  shape  is  each  ?  How 
many  square  inches  are  there  in  the  lower  face  of  the  6-in. 
cube  ?  How  many  cubic  inches  are  there  in  a  section 
of  it  1  in.  thick  ?  How  many  such  sections  are  there  ? 
How  many  cubic  inches  are  there,  then,  in  a  6-in.  cube  ? 


2.  How  many  cubic  inches  are  there  in  a  10-in.  cube  ? 
In  a  5-in.  cube  ?  In  a  12-in.  cube  ?  In  a  1-in.  cube  ? 
In  a  2-in.  cube  ?     In  a  4-in.  cube  ?     In  a  3-in.  cube  ? 

79 


80 


THE  STUDY  OF  SOLIDS 


3.  What  is  the  ratio  of  the  length  of  the  base  of  cube  a 
to  the  length  of  the  base  of  cube  b?  Of  the  width  of  a  to 
the  width  of  b?  .  Of  the  height  of  a  to  the  height  of  b? 
What  is  the  ratio  of  the  volume  of  a  to  the  volume  of  b? 

4.  What  is  the  ratio  of  a  cube  whose  length,  breadth, 
and  height  are  4  in.,  to  one  whose  dimensions  are  2  in.? 
Of  an  8-in.  cube  to  a  4-in.  cube? 

5.  Give  the  ratio  of : 

A  1-in.  cube  to  a  2-in.  cube. 
A  2-in.  cube  to  a  4-in.  cube. 

25-in.  cube  to  a  75-in.  cube. 

1-in.  cube  to  a  4-in.  cube. 

lj-in.  cube  to  a  3-in.  cube. 


A 
A 
A 
A 
A 
A 


6j~in.  cube  to  a  13-in.  cube. 


37fin. 


cube  to  a  12^-in.  cube. 


121 


£-m. 


A  56i-in. 


cube  to  a  25-in.  cube, 
cube  to  an  18|-in.  cube. 
A  25-in.  cube  to  a  5-in.  cube. 
6.   A    certain    box    measures    2    ft.   each  way,  inside. 
How  many  6-in.  cubes  of  wood  can  be  put  into  the  box? 


12 


< 

> 

c    12 

> 

b 

10 

4^ 

^4 

8\ 

""4 

7.   A  solid  which  has  two  bases  (or  ends)  which  are 
alike  and  parallel,  and  whose  sides  are  rectangles  (or  par- 


THE  STUDY  OF  SOLIDS  81 

allelograms)  is  called  a  prism,  a  and  b  are  square  prisms. 
Is  c  a  square  prism?  Is  a  cube  a  prism?  Is  a  cube  a 
square  prism?     A  solid  like  c  is  called  a  parallelepiped. 

8.  How  many  square  units  are  in  the  base  of  a?  Of 
b?  Of  c?  How  many  cubic  units  are  there  in  a?  In  b? 
Inc?    What  is  the  ratio  of  a  to  c ?    Of  a  to b?    Gib  toe? 

9.  What  is  the  ratio  of  the  length  of  the  base  of  a  to 
the  length  of  the  base  of  c?  Of  the  width  of  the  base  of 
a  to  the  width  of  the  base  of  c?  Of  the  base  of  a  to 
the  base  of  c?  Of  the  height  of  a  to  the  height  of  c?  If 
the  ratio  of  the  bases  of  a  and  c  is  f,  and  the  ratio  of  the 
heights  is  1,  what  is  the  ratio  of  the  volume  of  a  to  the 
volume  of  c?  Find  in  a  similar  way  the  ratio  of  a  to  b. 
Of  b  to  c. 

10.  Give  the  ratio  of  a  to  b : 

a  b 

Base  Height  Base  Height 

12  in.  by  8  in.,  15  in.  9  in.  by  4  in.,   10  in.  (f  x  f ) 

6  in.  by  7  in.,   16  in.  4  in.  by  14  in.,  32  in. 

121  in.  by  §\  in.,  16|  in.  18|  in.  by  121  in.,  25  in. 

31 J  ft.  by  331  ft.,  100  ft.  371  ft.  by  50  ft.,  1331  ft. 

8  ft.  by  9  ft.,   20  ft.  12  ft.  by  18  ft.,  30  ft. 

3  in.  by  3  in.,   6  in.  3  in.  by  6  in.,   12  in. 

18  in.  by  2  ft.,  5  yd.  12  in.  by  3  ft.,  10  yd. 

18f  in.  by  31  ft.,  6*  yd.  31*  in.  by  7  ft.,  18f  yd. 

40  ft.  by  40  yd.,  200  ft.  20  ft.  by  30  yd.,  66f  ft. 

8  ft.  by  8  ft.,   8  ft.  4  ft.  by  4  ft.,   4  ft. 

11.  If  a  chalk  box  measures  6  in.  x  4  in.  x  3  in.  outside, 
how  many  can  be  packed  in  a  box  4  ft.  x  3  ft.  x  2  ft.  ? 

MCN.   MENT.  AR.  6 


82 


THE  STUDY  OF  SOLIDS 


12.  How  many  times  as  much  air  is  there  in  a  room 
30  ft.  x  20  ft.  x  15  ft.,  as  in  a  room  20  ft.  x  25  ft.  x  18  ft.? 

13.  How  many  cubic  feet  of  water  are  there  in  a  tank 
16  ft.  long,  4  ft.  wide,  and  2  ft.  deep,  when  it  is  half  full? 
In  a  tank  32  ft.  long,  4  ft.  wide,  and  3  ft.  deep,  half  full? 

14.  How  many  3-in.  cubes  of  wood  can  be  put  into  a 
box  14  in.  long,  11  in.  wide,  and  10  in.  deep?  Can  the 
box  be  filled?     Why? 


a 


15.  Prisms  whose  bases  are  triangles  are  called  triangular 
prisms.  The  base  of  the  base  of  prism  a  is  5  units,  and  its 
altitude  4  units.  What  is  its  area?  If  its  area  is  10  square 
units,  how  many  cubic  units  are  there  in  a  section  1  unit 
thick,  cut  off  the  end  of  the  prism  ?  How  many  such  sec- 
tions are  there  in  a,  if  the  altitude  is  8  units  ?  How  many 
cubic  units  are  there  in  a?  How  many  cubic  units  are  there 
in  a  triangular  prism  8  units  high,  the  triangle  of  the  base 
having  a  base  of  5  units,  and  an  altitude  of  4  units? 

16.  What  is  the  volume  of  prism  5,  which  is  8  units 
high  and  whose  base  has  a  base  of  10  units,  and  an  altitude 
of  3  units  ? 


THE  STUDY  OF  SOLIDS  83 

17.  Give  the  volume  of  the  following  triangular  prisms : 
Height  18  in.,  base  of  base  5  in.,  altitude  of  base  4  in. 
Height*  20  in.,  base  of  base  12  in.,  altitude  of  base  6  in. 
Height  15  in.,  base  of  base  20  in.,  altitude  of  base  10  in. 
Height  100  in.,  base  of  base  50  in.,  altitude  of  base  4  in. 
Height  30  ft.,  base  of  base  121  ft.,  altitude  of  base  8  ft. 

18.  In  the  above  figures,  what  is  the  ratio  of  the  base  of 
the  base  of  prism  a,  to  the  base  of  the  base  of  prism  b? 
What  is  the  ratio  of  the  altitude  of  the  base  of  a  to  the 
altitude  of  the  base  of  b  ?  Of  the  area  of  the  base  of  a  to 
the  area  of  the  base  of  b  ?  What  is  the  ratio  of  the  height 
of  a  to  the  height  of  b  ?  If  the  ratio  of  the  bases  is  J,  and 
the  ratio  of  the  heights  is  1,  what  is  the  ratio  of  the 
volume  of  a  to  the  volume  of  b  ? 

19.  Give  the  ratio  of  prism  a  to  prism  b ;  of  c  to  d;  of 

e  to/;  of  g  to  h;  of  i  to  j. 

a.  Base  of  base    12  in.,  altitude      8  in.,  height    15  in. 

b.  Base  of  base      9  in.,  altitude      4  in.,  height    10  in. 

c.  Base  of  base      6  in.,  altitude      7  in.,  height    16  in. 

d.  Base  of  base      4  in.,  altitude    14  in.,  height    16  in. 

e.  Base  of  base  3  in.,  altitude  6  in.,  height  6  in. 
/.  Base  of  base  3  in.,  altitude  6  in.,  height  12  in. 
g.  Base  of  base  18|  in.,  altitude  3-|  ft.,  height  6 \  yd. 
h.  Base  of  base  31^  in.,  altitude  7  ft.,  height  18|  yd.- 
i.  Base  of  base  31^  in.,  altitude  33 J  in.,  height  100  in. 
y.  Base  of  base  37 J  in.,  altitude  50  in.,  height  133 J  in. 


84 


THE  STUDY  OF  SOLIDS 


20.   A  circular  solid  that  has  length  is  called  a  cylinder. 
A  round  pencil,  or  a  stove  pipe  is  a  cylinder.     What  is 


32 


24 


24 


the  shape  of  the  end  of  a  cylinder  if  cut  off  square  ?  How 
can  you  find  the  area  of  the  end  or  base  of  a  cylinder  if 
you  know  its  radius  ?     If  you  know  its  diameter  ? 

21.  If  the  diameter  of  the  base  of  cylinder  a  is  14  units, 
what  is  the  area  of  the  base  ?  If  there  are  154  square 
units  in  the  base,  how  many  cubic  units  are  there  in  a  sec- 
tion of  the  cylinder  1  unit  thick  ?  How  many  such  sec- 
tions are  there  if  the  height  of  the  cylinder  is  32  units  ? 
Are  there,  then,  32  x  154  cubic  units  in  cylinder  a  ? 

22.  The  volume  of  a  cylinder  is  represented  by  the 
number  of  units  in  the  height  times  the  number  of  square 


THE  STUDY  OF  SOLIDS 


85 


units  in  the  area  of  the  base.     What  is  the  volume  of  a 
cylinder  of  radius  7  in.  and  height  10  in.  ?    (1540  cu.  in.) 

Of  a  cylinder  of  radius  3^  in.  and  height  20  in.  ? 

Of  a  cylinder.of  radius     1  ft.  and  height     1  ft.  ? 

Of  a  cylinder  of  radius   1  yd.  and  height  2  yd.  ? 

23.  In  the  above  figures,  what  is  the  ratio  of  the  diameter 
of  a  to  the  diameter  of  c  ?  Of  the  area  of  the  base  of  a  to  the 
area  of  the  base  of  c  ?  (See  problem  37,  page  74.)  Of 
the  height  of  a  to  the  height  of  c  ?  If  the  ratio  of  the 
bases  is  4,  and  of  the  heights  is  |,  what  is  the  ratio  of 
the  volume  of  cylinder  a  to  the  volume  of  cylinder  c  ?  The 
ratio  of  one  cylinder  to  another  equals  the  product  of  the  ratio 
of  the  heights  by  the  square  of  the  ratio  of  the  diameters  or 
radii. 


24.    Give  the  ratio  of  cylinder  a  to  cylinder 

b: 

a 

Diameter  14    in. 

height    32    in. 

u 
Diameter    7    in. 

height  24    in. 

Diameter  18    ft. 

height    50    ft. 

Diameter  24    ft. 

height  75    ft. 

Diameter  12J  ft. 

height    83±  ft. 

Diameter  37|  ft. 

height  16|  ft. 

Diameter  56£  ft. 

height    41§yd. 

Diameter  31£  ft. 

height  75    yd. 

Diameter  16|  yd. 

,  height    66f  ft. 

Diameter  66|  yd.. 

,  height  83$  ft. 

Diameter  18f  rd., 

height    20    rd. 

Diameter  12|  rd. 

height  25    rd. 

Radius 

6    iii. 

height      2    ft. 

Diameter    3    in. 

height    4    ft. 

Radius 

12    in. 

height      4    ft. 

Diameter    2    ft. 

height    3    yd. 

Radius 

2£  ft. 

height  100  in. 

Diameter  12|  ft. 

height  20    in. 

Radius 

2i  in., 

height    40    ft. 

Diameter  18    in. 

height    5    ft. 

Radius 

5    ft. 

height    72    yd. 

Diameter  30    ft. 

height    2    yd, 

Radius 

32    in. 

height    45    in. 

Diameter  40    in.. 

,  height  72    in. 

25.   How  many  cubic  feet  of  water  will  a  cylindrical 
tank  2  ft.  in  diameter  and  7  ft.  high  hold  ? 


86 


THE  STUDY  OF  SOLIDS 


26.  A  cylindrical  pail  8  in.  in  diameter  and  12  in.  deep 
holds  how  many  times  as  much  as  a  pail  6  in.  in  diameter 
and  8  in.  deep  ? 

27.  How  many  gallons  will  a  cylindrical  cistern  15  ft. 
deep  hold  if  the  bottom  has  an  area  of  30  sq.  ft.,  and  it  is 
half  full  ?     (See  problem  42,  page  60.) 


28.  A  solid  that  has  a  base  of  any  number  of  sides,  and 
whose  other  surfaces  are  triangles  meeting  at  a  common 
point  at  the  top  (the  apex  of  the  pyramid)  is  a  pyramid. 
The  altitude  is  the  height,  or  usually  the  shortest  distance 
from  the  apex  to  the  base.  Figure  a  represents  a  pyramid 
with  a  rectangular  base.  Figures  b  and  e  represent  rect- 
angular pyramids  inside  of  prisms  of  base  and  height  equal 
to  those  of  the  pyramids.  Figure  d  represents  a  triangular 
pyramid  inside  a  prism  of  equal  base  and  height.  The 
volume  of  a  pyramid  is  represented  by  \  the  product  of 
the  number  of  square  units  in  its  base  by  the  number  of 
units  in  its  altitude.     What  is  the  volume  of  pyramid  a, 


THE  STUDY  OF  SOLIDS  87 

if  its  base  is  3  units  by  2  units,  and  its  altitude  is  8  units? 
Of  pyramid  b,  if  its  base  is  4  units  by  3  units,  and  its 
altitude  is  8  units  ?  Of  pyramid  <?,  base  3  units  by  6  units, 
and  altitude  10  units?  Of  pyramid  d,  base  of  base  3  units, 
altitude  of  base  4  units,  altitude  of  pyramid  12  units  ? 

29.  Give  the  volume  of  the  following  pyramids  : 
Base,  a  rectangle,  8  in.  by  6  in.,  height  20  in. 
Base,  a  rectangle,  3  ft.  by  5  ft.,  height  15  ft. 
Base,  a  rectangle,  1  yd.  by  3  ft.,  height  25  ft. 

Base,  a  triangle,  base  18  ft.,  altitude  10  ft.;  height  20  ft. 
Base,  a  triangle,  base  8  yd.,  altitude  6  yd.;  height  8  yd. 
Base,  a  triangle,  base  6  in.,  altitude  5  in.;  height  8  in. 

30.  In  the  above  figure,  what  is  the  ratio  of  the  base  of 
pyramid  a  to  the  base  of  pyramid  b  ?  Of  the  height  of  a 
to  the  height  of  b  ?  If  the  ratio  of  the  bases  is  J  unit,  and 
of  the  heights  is  1  unit,  what  is  the  ratio  of  the  base  x  height 
of  a  to  the  base  x  height  of  b  ?  Of  £  base  x  height  of  a  to 
J  base  x  height  of  b  ?  Of  the  volume  of  a  to  the  volume 
of  b  ?  What  is  the  ratio  of  the  base  of  b  to  the  base  of  c  ? 
Of  the  height  of  b  to  the  height  of  c  ?  Of  the  volume  of  b 
to  the  volume  of  c  ?  (y8^)  Why  is  it  not  necessary  to  con- 
sider the  ^  ?     (See  problem  28.) 

31.  In  the  following  rectangular  pyramids  give  the 
ratio  of  a  to  b  : 


a 
Base 
3  in.  x  4  in. 

Height 

8  in. 

b 
Base 
6  in.  x  3  in. 

Height 
10  in. 

25in.xl6f  in. 

62i  in. 

3l£  in.  x  66|  in. 

37|  in. 

L8fft.x3Hft. 

33i  ft. 

25  ft.  x  37i  ft. 

41|  ft. 

8  yd.  x  4  yd. 
LOO  ft.  x  100  yd. 

10  yd. 
100  in. 

6  yd.  x  8  yd. 
200  ft.  x  200  yd. 

15  yd. 
200  in. 

88 


THE  STUDY  OF  SOLIDS 


32.    In  the  following  triangular  pyramids  give  the  ratio 
of  a  to  b. 


a 
Base 

Height 

b 
Base 

Height 

3  in.  base      4  in.  alt. 

12  in. 

4  in.  base 

6  in.  alt. 

Sin. 

18|  in.  base     8|  in.  alt. 

31iin. 

12|  in.  base 

16$  in.  alt. 

18|  in. 

5  ft.  base      4  ft.  alt. 

12  ft. 

6  ft.  base 

10  ft.  alt. 

16  ft. 

8  yd.  base     12  ft.  alt. 

150  in. 

6  yd.  base 

16  ft.  alt. 

100  in. 

1  yd.  base       1  ft.  alt. 

1yd. 

1  ft.  base 

1  yd.  alt. 

1  ft. 

33.  A  solid  having  a  circular  base,  and  tapering  to  a 
point,  is  a  cone.  The  cone  bears  the  same  relation  to 
a  cylinder  that  a  pyramid  does  to  a  prism.  The  volume 
of  a  cone  is  represented  by  J  of  the  product  of  the  area  of 
its  base  by  its  height.  What  is  the  volume  of  a  cone  whose 
base  is  20  sq.  in.,  and  whose  height  is  27  in.  ?  Of  a  cone 
whose  height  is  30  in.,  and  whose  base  has  a  radius  of 
7  in.  ?  (1540  cu.  in.)  Of  a  cone  whose  height  is  21  ft., 
and  whose  base  has  a  radius  of  1  ft.  ? 


(   .  ) 

3 

, ^ 

a\        \ 

2 

b 

L-—- 

^^  i  7 

34.    Cones  a  and  b  are  represented  inside  of  cylinders 
of  height  and  base  equal  to  those  of  the  cone.     What  is 


THE  STUDY  OF  SOLIDS 


89 


the  ratio  of  the  base  of  cone  a  to  the  base  of  cone  6,  if  the 
diameters  are  2  and  1  ?  (See  problem  23,  page  85.)  What 
is  the  ratio  of  the  height  of  a  to  the  height  of  b  ?  Of  the 
product  of  the  height  by  the  base  of  a,  to  the  product 
of  the  height  by  the  base  of  b  ?  Of  ^  of  base  x  height  of 
a  to  ^  of  base  x  height  of  J?  Why  is  it  not  necessary 
to  consider  the  ^  in  getting  the  ratio  of  one  cone  to 
another  ? 


35.    Give  the  ratio  of  cone  a  to  cone  b : 


Radius  of  Base 

Height 

Radius  of  Base 

Height 

10  in. 

20  in. 

5  in. 

10  in. 

12  in. 

16  in. 

18  in. 

20  in. 

18|  in. 

41f  m. 

25  in. 

75  in. 

Diameter  of  Base 

Height 

Diameter  of  Base 

Height 

50  in. 

6^  ft. 

75  in. 

121  ^ 

25  ft. 

25  ft. 

25  yd. 

25  yd. 

Circumference  of  Base    Height    Circumference  of  Base    Height 
60  in.  28  in.  80  in.  63  in. 


36.  What  is  the  ratio  of  a  cone  to  a  cylinder  of  the  same 
base  and  height  ?  Of  a  pyramid  to  a  prism  of  the  same 
base  and  height  ? 

37.  How  many  times  will  a  conical  dipper  2  in.  in 
diameter  at  the  top,  and  4  in.  deep,  have  to  be  dipped 
full  of  milk  to  fill  a  cylindrical  pail  6  in.  in  diameter 
and  8  in.  deep  ? 

What  is  the  ratio  of  the  cone  to  a  cylinder  of  the  same  base  and 
height  ?  Of  this  cylinder  to  the  pail  ?  Of  the  dipper  to  the  pail  ?  Of 
the  pail  to  the  dipper  ? 


90  THE  STUDY  OF  SOLIDS 

38.  Which  is  greater  in  volume  and  how  much,  a  cone 
6  in.  high,  whose  base  has  a  radius  of  1  ft.,  or  a  square 
pyramid  6  in.  high,  whose  base  is  2  ft.  square  ? 

39.  Which  is  greater  in  volume  and  how  much,  a  cylin- 
der 8  in.  high  and  2  in.  in  diameter,  or  a  prism  8  in.  high 
and  2  in.  square  ? 

40.  What  is  the  diameter  of  a  cylindrical  jar  6  in.  deep 
that  holds  one  gallon  ?     Of  a  2  gal.  jar  12  in.  deep  ? 

41.  What  is  the  height  of  a  square  pyramid  whose  base 
is  1  ft.  square  and  whose  volume  is  1440  cu.  in.  ?  Of  a 
cone  whose  volume  is  44  cu.  in.  and  base  2  in.  in  diameter  ? 

42.  What  is  the  radius  of  the  base  of  a  cone  whose 
volume  is  22  cu.  ft.  and  height  21  ft.  ?  Of  one  whose 
volume  is  1540  cu.  in.  and  height  30  in.  ? 

43.  How  many  boxes  6|-  in.  x  5  in.  x  4  in.  can  be  packed 
into  a  box  1  ft.  7  j-  in.  by  1  ft.  3  in.  by  1  ft.  4  in.  ? 

44.  A  sphere  is  a  solid  that  has  all  points  in  its  sur- 
face equally  distant  from  a  point  within  called  the  center. 
A  ball  is  a  sphere.  The  radius  is  the  distance  from  the 
center  to  any  point  on  the  surface.  The  diameter  is  twice 
the  radius,  or  the  distance  from  any  point  on  the  surface 
to  the  point  of  the  surface  farthest  away.  The  volume  of 
a  sphere  is  represented  by  |x^2-xrxrxr,  that  is,  ^ 7rr3. 
What  is  the  volume  of  a  sphere  of  1  in.  radius?  (|  x -2T2-  X 
lxlxl  =  ?) 

45.  If  the  radius  of  sphere  a  is  1,  and  of  sphere  b  is  2, 
the  volume  of  a  is  represented  hj  |  x-2T2-xl  xl  x  1;  and 
of  b  by  |  x  -2y2-  x  2  x  2  x  2.  What  is  the  ratio  of  the  volume 
of  a  to  the  volume  of  b?     (|)     What  is  the  ratio  of  a 


THE  STUDY  OF  SOLIDS  91 

sphere  of  2  in.  radius  to  one  of  4  in.  radius  ?  .(J  x  \  X  J) 
Of  one  of  3  in  radius  to  one  of  6  in  radius  ?  Of  one  of 
diameter  1  in.  to  one  of  diameter  2  in.  ?  The  ratio  of  one 
sphere  to  another  equals  the  cube  of  the  ratio  of  the  radii  or 
of  the  diameters.  Why  is  it  not  necessary  to  consider 
either  the  J  or  the  -^2-  ? 

46.  Give  the  ratio  of  a  sphere : 

Of  radius  4  in.  to  one  of  radius  8  in. 

Of  radius  3^  in.  to  one  of  radius  7  in. 

Of  radius  18|  ft.  to  one  of  radius  37 1  ft. 

Of  radius  1  in.  to  one  of  radius  3  in. 

Of  radius  6|  ft.  to  one  of  radius  18|  ft. 

Of  radius  3  ft.  to  one  of  radius  4  ft.  (f  x  f  x  f ). 

Of  radius  2|  ft.  to  one  of  radius  |  ft. 

Of  diameter  25  ft.  to  one  of  diameter  37 J  ft. 

Of  diameter  25  ft.  to  one  of  diameter  83J  ft. 

Of  diameter  5|  ft.  to  one  of  diameter  2|  ft. 


SECTION  XI 

REVIEW  OF  MEASUREMENTS 

SUGGESTIONS 

If  the  pupil  encounters  a  problem  he  cannot  solve,  he  should 
return  to  the  development  of  the  idea  as  first  given  on  preceding 
pages,  diagram  the  problem,  think  it  through,  and  study  to  express 
his  ideas  clearly  in  simple  scientific  terms.  When  a  technical  word, 
representing  an  idea  fully  developed,  is  in  the  learner's  mind  he 
should  use  it. 

1.  Give  the  ratio  of : 

A  1-in.  square  to  a  2-in.  square. 
A  6-in.  square  to  a  9-in.  square. 
An  18f-in.  square  to  a  25-in.  square. 
A  1-in.  circle  to  a  2-in.  circle. 
A  6J-in.  circle  to  a  9|-in.  circle. 
A  31^-in.  circle  to  a  50-in.  circle. 
A  1-in.  cube  to  a  2-in.  cube. 
A  21-in.  cube  to  a  28-in.  cube. 
An  83^-in.  cube  to  a  33^-in.  cube. 
A  1-in.  sphere  to  a  2-in.  sphere. 
A  41|-in.  sphere  to  a  33l-in.  sphere. 
A  75-in.  sphere  to  a  100-in.  sphere. 

2.  What  is  the  length  of  the  side  of  a  square  that  con- 
tains |  of  the  area  of  a  4-in.  square  ? 

If  £  is  the  ratio  of  the  areas,  what  is  ratio  of  the  sides?  The  side 
of  the  required  square  is  then  what  part  of  4  in.  ? 

92 


BEVIEW  OF  MEASUREMENTS  93 

3.  What  is  the  length  of  the  side  of  a  square  that 
equals  ^  of  the  area  of  a  25-in.  square  ?  f$  of  the  area 
of  a  31J-in.  square  ?  Jf  of  the  area  of  a  35-in.  square  ? 
-^  of  the  area  of  a  21-in.  square  ? 

4.  What  is  the  diameter  of  a  circle  that  equals  -^  of 
the  area  of  a  25-in.  circle  ?  ||  of  the  area  of  a  28-in. 
circle  ?  *£•  of  the  area  of  a  12-in.  circle  ?  §  J  of  the  area 
of  a  41  J-in.  circle  ? 

5.  What  is  the  length  of  a  side  of  a  cube  which  equals 
28y  of  the  volume  of  a  12-in.  cube  ?  |J  of  the  volume  of  a 
25-in.  cube  ?  8  times  the  volume  of  a  12J-in.  cube  ?  i|£ 
of  the  volume  of  a  10-in.  cube  ? 

6.  What  is  the  diameter  of  a  sphere  that  equals  27  of 
the  volume  of  a  15-in.  sphere  ?  -^  of  the  volume  of  a 
12l-in.  sphere  ?  ±^-  of  the  volume  of  a  33|-in.  sphere  ? 
J£  of  the  volume  of  a  75-in.  sphere  ? 

7.  How  many  lj-in.  cubes  can  be  put  into  a  box  1\  in. 
each  way  ? 

8.  How  many  steps  2  ft.  6  in.  long  are  taken  in  walk- 
ing 5  rd.  ? 

What  is  the  ratio  of  5  ft.  to  1\  ft.  ?    Then  of  5  rd.  to  2|  ft.  ? 

9.  How  many  times  must  a  pail  holding  2£  qt.  be 
filled  with  water  in  order  to  fill  a  10-gal.  can  ? 

10.  In  digging  a  ditch  40  ft.  long,  3  ft.  wide,  and  6  ft. 
deep,  how  many  cubic  yards  of  dirt  are  removed  ? 

11.  A's  land  measures  40  rd.  on  each  side,  and  B's  60 
rd.  on  each  side.  What  is  the  ratio  of  B's  land  to  A's  ? 
If  A's  is  worth  %  4000,  how  much  is  B's  worth  at  the  same 
rate  ? 


94  REVIEW  OF  MEASUREMENTS 

12.  A's  land  is  1  mi.  square,  B's  is  160  rd.  square.  If 
B's  land  is  worth  $625,  how  much  is  A's  worth  at  the 
same  rate? 

13.  A's  land  is  80  rd.  by  160  rd.,  B's  240  rd.  by  120  rd. 
If  A's  land  is  worth  $8000,  how  much  is  B's  worth  at  the 
same  rate  ? 

14.  A  city  block,  trapezoidal  in  form,  measures  300  ft. 
and  200  ft.  on  its  parallel  sides.  The  distance  between 
these  two  sides  is  250  ft.     What  is  its  area  in  square  feet  ? 

15.  A  horse  is  tied  to  a  stake  by  a  rope  3J  yd.  long. 
Over  how  many  square  yards  of  ground  can  he  feed  ? 

16.  If  a  pipe  6  in.  in  diameter  carries  80  gal.  per 
minute,  how  many  gallons  per  minute  will  a  9-in.  pipe 
carry,  the  water  running  at  the  same  speed  ?  A  12-in. 
pipe  ?     A  15-in.  pipe  ? 

17.  If  a  box  4  ft.  x  3  ft.  x  1|  ft.  full  of  sand  weighs 
2000  lb.,  how  much  will  a  box  6  ft.  x  4  ft.  x  3  ft.  weigh, 
full  of  the  same  sand  ? 

18.  A  well  70  ft.  deep  was  bored  with  a  2-ft.  auger. 
How  many  cubic  feet  of  earth  were  removed  ? 

19.  How  many  cubic  feet  are  in  the  loft  of  a  barn  30  ft. 
long,  the  triangular  ends  having  a  base  of  20  ft.  and  an 
altitude  of  15  ft.  ? 

20.  What  is  the  ratio  of  1  rd.   1  yd.   1  ft.  4  in.  to 

2  ft.  1  in.  ? 

21.  How  many  feet  of  lumber  are  necessary  to  cover, 
with  inch  boards,  the  walls  of  a  house  20  ft.  x  20  ft.  and 
15  ft.  high? 


REVIEW  OF  MEASUREMENTS  95 

I 

22.  How  many  cubic  feet  of  wood  are  there  in  a  log 
14  ft.  long  and  2  ft.  in  diameter  ? 

23.  A  piece  of  metal  roofing  is  cut  so  that  two  sides  are 
parallel,  and  the  distance  between  them  is  10  ft.  If  the 
two  parallel  sides  measure  12  ft.  and  8  ft.,  what  does  the 
piece  weigh  at  the  rate  of  1J  lb.  per  square  foot  ? 

24.  If  a  3|-in.  iron  ball  weighs  16  lb.,  how  much  will 
a  5J-in.  iron  ball  weigh  ? 

25.  If  a  sheet  metal  circle  20  in.  in  diameter'  weighs 
5  lb.,  how  much  will  a  30-in.  circle  of  the  same  metal 
weigh  ? 

26.  The  speed  being  the  same,  how  much  water  will  a 
3-in.  pipe  carry  in  an  hour  if  a  f-in.  pipe  carries  50  gal.  ? 

27.  Of  two  cylinders  of  wood  of  equal  length  one 
measures  18  j  in.  around,  and  the  other  25  in.  If  the  first 
weighs  27  lb.,  how  much  does  the  other  weigh? 

28.  How  many  J-in.  pipes  could  carry  the  same  amount 
of  water  at  the  same  rate  as  two  2-in.  pipes  ? 

29.  If  a  block  of  ice  3  ft.  x  4  ft.  x  2J  ft.  weighs  1800  lb., 
what  is  the  weight  of  a  piece  2  ft.  x  6  ft.  x  5  ft.  ? 

30.  If  the  price  asked  for  panes  of  glass  12  in.  x  15  in.  is 
27  cjts.  each,  how  much  should  a  pane  10  in.  x  16  in.  cost? 

31.  How  many  times  as  much*  land  can  a  horse  feed 
over  when  tied  with  a  30-ft.  rope  as  when  tied  with  a 
25-ft.  rope  ? 

32.  How  many  blocks  2  in.  x  2  in.  x  4  in.  can  be  packed 
in  a  box  2  ft.  x  2  ft.  x  4  ft.,  measured  inside  ? 


96  REVIEW  OF  MEASUREMENTS 

33.  About  how  many  gallons  of  water  can  be  put  into  a 
tank  measuring  7J  ft.  x  4  ft.  x  3  ft.  ? 

34.  Of  two  cylinders  of  equal  diameter,  one  is  f  as  long 
as  the  other.     What  is  the  ratio  of  the  second  to  the  first? 

35.  What  is  the  capacity  in  cubic  inches  of  a  cone- 
shaped  dipper  6  in.  deep  and  7  in.  in  diameter  at  the  top  ? 

36.  What  is  the  radius  of  the  base  of  a  cone  whose 
volume  is  22  cu.  ft.,  and  whose  height  is  14  ft.  ? 

37.  Which  is  greater,  and  how  much,  a  square  44  in. 
around  or  a  circle  44  in.  around  ? 

38.  Which  will  hold  more,  and  how  much,  a  square  pail 
10  in.  deep  and  44  in.  around  or  a  cylindrical  pail  of  the 
same  dimensions? 

39.  If  a  carriage  wheel  turns  15  times  in  running  10  rd., 
what  is  its  diameter  ? 

40.  If  a  tin  fruit  can  4J  in.  high  and  4  in.  in  diameter 
holds  a  quart,  what  is  the  height  of  a  gallon  can  8  in.  in 
diameter?  sWhat  would  be  the  height  of  a  gallon  can 
4  in.  in  diameter  ? 

41.  What  must  be  the  dimensions  of  a  box  in  order  to 
hold  exactly  420  blocks,  each  1J  in.  x  2  in.  x4  in.  ? 

What  are  the  factors  of  420  that  will  contain  \\  in.,  2  in.,  and  4  in. 
respectively  ? 

42."  What  must  be  the  dimensions  of  a  box  to  hold 
exactly  36  blocks,  each  2  in.  x  1  in.  x  3  in.  ? 

43.  At  $  .11  per  pound,  what  is  the  value  of  a  cheese  14 
in.  in  diameter  and  10  in.  high,  weighing  1  lb.  to  14  cu.  in.  ? 


REVIEW  OF  MEASUREMENTS  97 

44.  If  a  cylinder  2  ft.  in  diameter  and  7  ft.  high  weighs 
2100  lb.,  what  is  the  weight  of  one  of  the  same  material 
5^  ft.  in  diameter  and  3  ft.  high  ? 

45.  If  a  rectangular  block  of  wood  8  ft.  by  18  in.  by 
14  in.  weighs  42  lb.,  what  is  the  weight  of  a  block  of  the 
same  wood  12  ft.  by  24  in.  by  12  in  ? 

46.  What  is  the  ratio  of  the  value  of  a  6^-in.  ball  of 
gold  to  a  12^-in.  ball  of  silver,  if  the  gold  is  worth  24 
times  as  much  as  the  silver,  per  cubic  inch  ? 

47.  How  many  revolutions  does  a  28-in.  bicycle  wheel 
make  in  going  19  yd.  1  ft.  8  in.  ? 

48.  A  vat  is  built  in  the  form  of  an  inverted  pyramid. 
The  top  is  10  ft.  square,  and  the  depth  is  10  ft.  About 
how  many  gallons  does  it  hold  ? 

How  many  would  it  hold  if  cubical  instead  of  pyramidal? 

49.  Which  is  larger  and  how  much,  a  two-inch  cube  or 
a  two-inch  sphere  ? 

50.  A  circular  inclosure  contains  154  sq.  rd.  How 
many  rods  in  diameter  is  it  ?  What  would  be  the  diame- 
ter of  a  similar  inclosure  containing  4  times  154  sq.  rd.  ? 
Of  one  containing  38J  sq.  rd.  ? 

51.  What  is  the  ratio  of  A's  land  to  B's,  if  A's  is  80  rd. 
long  and  60  rd.  wide,  and  B's  is  60  rd.  long  and  60  rd.  wide  ? 

52.  How  many  2-inch  cubes  would  weigh  as  much  as  an 
8-inch  cube  of  the  same  material  ? 

53.  How  many  feet  of  inch  boards  are  needed  to  build  a 
tight  board  fence  6  ft.  high  and  1000  ft.  long  ? 

54.  If  a  round  iron  plate  ^  in.  thick  weighs  10  lb.,  what 
does  a  round  iron  plate  5  times  the  diameter  of  the  first 
and  1  in.  thick  weigh  ? 

MCN.  MENT.  AR.  —  7 


98  REVIEW  OF  MEASUREMENTS 

55.  What  is  the  value  of  50  J-in.  boards  12  ft.  long  and 
6  in.  wide,  at  $ 20  per  M.  ? 

56.  At  $2.50  per  M.  for  shingles  and  $.50  per  M.  for 
laying  them,  what  will  be  the  cost  of  shingling  a  hip  roof 
which  has  two  equal  triangular  portions,  base  20  ft.  and 
altitude  30  ft.,  and  two  equal  trapezoidal  portions,  bases 
20  ft.  and  10  ft.,  and  altitude  30  ft.  ? 

57.  How  many  feet  are  traversed  at  each  complete  revo- 
lution of  the  pedals  by  a  bicycle  having  28-in.  wheels,  if 
the  front  sprocket  has  27  teeth,  and  the  rear  one  9  ? 

How  raany  turns  does  the  rear  wheel  make  at  each  turn  of  the 
pedals  ? 

58.  How  many  bushels  of  grain  will  a  bin  8  f t.  x  5  ft.  x 
i  ft.  hold? 

59.  Find  the  ratio  of  areas  or  volumes  : 
A  16|-in.  circle  to  a  25-in.  circle. 

A  rectangle  6  ft.  x  4  ft.  to  one  1^  ft.  x  3  ft. 

A  16§-in.  cube  to  a  25-in.  cube. 

A  prism  4  in.  x  6  in.  x  4  in.  to  one  2  in.  x  3  in.  x  2  in. 

A  4-in.  square  to  a  6|-in.  square. 

An  831-f t.  sphere  to  a  66|-f t.  sphere. 

A  3-in.  sphere  to  a  4J-in.  sphere. 

A  2|-yd.  cube  to  a  1-rd.  cube. 

Any  circle  to  a  circle  of  twice  its  diameter. 

Any  sphere  to  one  of  twice  its  radius. 

Any  circle  to  one  of  3  times  its  circumference. 

Any  cube  to  one  of  2|  times  its  length. 

Any  sphere  to  one  of  4  times  its  diameter. 


REVIEW  OF  MEASUREMENTS  99 

60.  Find  the  ratio  of.  a  cone  whose  base  is  3f  in.  in 
diameter,  and  whose  height  is  6  in.,  to  one  whose  base  is 
14  in.  in  diameter,  and  height  3  in. 

61.  Find  the  ratio  of  a  triangle  of  base  33^  in.,  altitude 
16|  in.,  to  one  of  base  66|  in.,  altitude  33 J  in. 

62.  Find  the  ratio  of  a  pyramid  of  rectangular  base 
16|  in.  x  18f  in.,  and  10  ft.  high,  to  one  of  rectangular 
base  25  in.  x  25  in.,  and  20  ft.  high. 

63.  What  is  the  capacity  in  gallons  of  a  jar  14  in.  in 
diameter  and  10  in.  deep  ? 

How  deep  must  it  be  to  hold  1  gal.? 

64.  How  many  cubic  inches  of  wood  are  removed  in 
boring  through  a  timber  14  in.  thick  with  a  2-in.  auger? 

65.  What  is  the  diameter  of  a  sphere  whose  volume  is 
|f  of  a  16§-ft.  sphere  ? 

66.  How  many  times  can  a  cylindrical  pail  7  in.  in 
diameter  and  12  in.  deep  be  filled  from  a  pail  14  in.  in 
diameter  and  16  in.  deep  ? 

67.  What  is  the  ratio  of  the  side  of  a  cube  whose  vol- 
ume is  125  cu.  ft.  to  the  side  of  a  cube  whose  volume  is 
64  cu.  ft.  ?     27  cu.  ft.  ?     8  cu.  ft,  ? 

68.  Which  is  greater,  and  how  much,  a  rectangular 
prism  2  ft.  x  3  ft.  and  4  ft.  high,  or  a  triangular  prism, 
base  of  base  4  ft.,  altitude  of  base  3  ft.,  and  5  ft.  high  ? 

69.  Think  of  a  sphere  inside  a  2-in.  cube,  touching  all 
six  faces.     Which  is  larger,  and  how  much  ? 

70.  How  many  bushels  of  wheat  will  a  bin  8  ft.  x  10  ft. 
x  4  ft.  hold ?    A  bin  5  ft.  x  3  ft.  x  6  ft.? 


100  REVIEW  OF  MEASUREMENTS 

71.  About  how  many  gallons  will  a  tank  15  ft.  by  6  ft. 
by  4  ft.  hold  ? 

How  many  gallons  are  there  in  15  x  1  x  1  cu.  ft.  ?  In  6  x  4  x  that 
space  ? 

72.  What  must  be  the  depth  of  a  cylindrical  jar  7  in. 
in  diameter  in  order  to  hold  a  gallon  (231  cu.  in.)? 

73.  At  $10  per  M.,  how  much  are  twelve  2  x  6's  20  ft. 
long,  and  nine  4  x  4's  12  ft.  long  worth  ? 

74.  How  many  shingles  are  needed  for  a  roof  40  ft.  by 
20  ft.  ?     How  much  will  they  cost  at  $3  per  M.  ? 

75.  A  certain  hip  roof  has  two  equal  triangular  portions, 
base  30  ft.  and  altitude  20  ft.,  and  two  equal  trapezoidal 
portions,  bases  30  ft.  and  10  ft.,  and  altitude  15  ft.  How 
much  will  the  shingles  cost  for  the  roof  at  $2.50  per  M.  ? 


SECTION  XII 

INTRODUCING  THE  IDEAS  OF  ANALYSIS  AND  MENTAL 
ALGEBRA  IN  SOLVING  PROBLEMS  AND  FINDING 
RELATIONS 

SUGGESTIONS 

a.  Analysis  presents  very  lew  new  ideas.  This  chapter  contains 
problems  which  may  in  many  instances  be  solved  by  several  methods. 
The  purpose  is  to  cultivate  "  common  sense  "  as  applied  to  the  solu- 
tion of  problems  in  arithmetic.  The  best  work  is  characterized  by 
terse  statements  of  relations  in  clear  grammatical  sentences.  In- 
volved complex  statements  obscure  thought  processes.  The  learner 
should  acquire  the  habit  of  arriving  at  true  results ;  and  he  should 
feel  that  the  clearness  and  accuracy  of  his  sentences  measure  the 
growth  and  development  of  his  mathematical  insight. 

b.  Beginning  with  problem  61  of  this  section,  by  easy  steps  the 
learner  is  inducted  into  a  new  method  of  solving  problems.  Mental 
algebra  is  a  very  attractive  field.  The  power  to  apprehend  abstract 
relations,  to  hold  them  in  the  mind,  and  to  group  them  so  as  to  bring 
the  relations  into  a  comprehended  whole  is  agreeable  and  profitable 
work. 

c.  Should  there  be  any  pupils  in  the  class  who  do  not  see  clearly 
the  force  of  transposition,  the  teacher  will  find  that  the  conception  of 
change  of  signs  as  developed  in  Milne's  Elementary  Algebra,  or  any 
other  simple  statement  of  the  idea,  will  help.  Guidance  at  this  point, 
if  needed,  will  inspire  confidence  in  ability  to  do. 

l.  If  |  of  A's  money  equals  f  of  B's,  and  both  have  $  90, 
how  much  has  each  ? 

All  of  A's  money  is  what  part  of  f  of  A's  money? 

Then,  since  f  of  A's  =  f  of  B's,  all  of  A's  money  =  what  part  of  f  of 
B's?    All  of  A's  is  what  part  of  B's  ?    Then,  if  A's  =  £  of  B's,  A's  and 

101 


102  ANALYSIS  AND  MENTAL  ALGEBRA 

B's  together  =  how  many  4ths  of  B's  ?     If  $  90  is  f  of  B's,  how  many- 
dollars  has  B  ?    How  many  has  A  ? 

2.  If  A  and  B  together  have  $  92,  and  f  of  A's  money 
equals  |-  0f  B's,  how  much  has  each  ? 

Ratio  of  A's  to  f  of  A's  ?  Of  A's  to  f  of  B's  ?  Of  A's  to  B's  ?  Of 
$  92  to  B's  ?    B's,  then  =  what  part  of  $  92  ?    B's  =  ?    A's  =  ? 

3.  A  and  B  engage  in  business  with  a  joint  capital  of 
$3800.  If  f  of  A's  capital  equals  f  of  B's,  how  much 
does  each  invest  ? 

4.  John  and  James  started  from  the  same  place  and 
walked  in  opposite  directions.  At  the  end  of  a  certain 
time  they  were  480  rd.  apart.  If  f  of  the  distance  John 
walked  equals  f  of  the  distance  James  walked,  how  far 
did  each  walk  ? 

5.  y9T  of  one  number  equals  Jf  of  another  number.  If 
their  sum  is  270,  what  are  the  numbers  ?  If  the  sum  is 
360?     135?     225? 

6.  The  sum  of  two  numbers  is  12,  and  f  of  the  first 
equals  f  of  the  second.     What  are  the  numbers  ? 

7.  What  time  is  it  when  |  of  the  time  past  noon  equals 
|  of  the  time  to  midnight  ?  When  |  of  the  time  to  mid- 
night equals  f  of  the  time  past  noon  ? 

8.  Two  men  are  105  rd.  apart.  They  walk  toward 
each  other  till  they  meet,  at  such  rates  that  £  of  the  dis- 
tance the  first  travels  equals  f  of  the  distance  the  second 
travels.     How  far  does  each  walk  ? 

9.  A  and  B  are  900  rd.  apart.  They  travel  toward 
each  other  till  they  meet,  A  at  the  rate  of  20  rd.  per 


ANALYSIS  AND  MENTAL  ALGEBRA  103 

minute  and  B  at  the  rate  of  25  rd.  per  minute.     How 
many  rods  does  each  travel  before  they  meet  ? 

Both  travel  how  many  rods  per  minute  ?  If  the  whole  distance  is  900 
rd.,  how  many  minutes  do  they  travel?     How  far  does  A  travel?    B? 

10.  Two  trains  480  mi.  apart  run  toward  each  other  till 
they  meet,  the  first  running  30  mi.  per  hour  and  the  other 
50  mi.  per  hour.  How  far  does  each  travel  before  they 
pass? 

11.  Two  horsemen  start  from  the  same  place  at  -the  same 
time,  going  in  the  same  direction.  The  first  travels  10  mi. 
per  hour,  and  the  second  7 J  mi.  per  hour.  How  far  has 
each  traveled  when  they  are  35  mi.  apart  ? 

How  far  apart  do  they  get  in  1  hr.  ? 

12.  A  and  B  start  together  from  a  certain  point,  going 
in  the  same  direction.  B  walks  at  the  rate  of  3|  mi.  per 
hour,  and  A  at  the  rate  of  4|  mi.  per  hour.  How  far  has 
each  traveled  when  they  are  12  mi.  apart  ? 

13.  A  mounted  messenger  is  sent,  traveling  at  the  rate 
of  8  mi.  per  hour.  Three  hours  later  a  second  one  is  sent 
after  him,  traveling  at  the  rate  of  10  mi.  per  hour.  How 
far  did  the  latter  travel  before  overtaking  the  former  ? 

14.  A  man  started  from  home  at  8  A.M.  and  traveled 
horseback  at  the  rate  of  1  mi.  per  hour  to  a  certain  place. 
Then  selling  his  horse  he  walked  back  at  the  rate  of  3^  mi. 
per  hour,  reaching  home  at  12.30  p.m.  How  far  did  he 
ride? 

What  is  the  ratio  of  the  time  he  walked  to  the  time  he  rode  ?  Then 
4|  hr.  must  equal  how  many  times  the  time  he  rode  ?  How  long  did 
he  ride  ?     How  far,  at  7  mi.  per  hour  ? 


104  ANALYSIS  AND  MENTAL  ALGEBRA 

15.  A  boy  starts  from  home  at  7  a.m.  on  his  bicycle, 
riding  at  the  rate  of  10  mi.,  per  hour.  After  riding  a  cer- 
tain distance  he  meets  with  an  accident  and  has  to  walk 
to  his  destination,  going  at  the  rate  of  3  mi.  per  hour,  and 
reaches  the  end  of  his  journey  at  12.20  p.m.  If  the  whole 
distance  is  30  mi.,  how  far  does  he  ride,  and  how  far  does 
he  walk  ? 

16.  A  boy  walks  along  a  road,  followed  by  a  man  10  rd. 
away.  They  step  at  the  same  rate,  but  the  man's  steps 
are  2  ft.  6  in.  long,  and  the  boy's  2  ft.  How  far  must  the 
man  travel  before  he  overtakes  the  boy  ? 

How  much  does  the  man  gain  at  each  step?  How  many  steps 
must  he  take  to  gain  1  rd.  ?  How  many  steps  does  he  take  in  gaining 
10  rd.  ?     How  many  rods  does  he  gain  in  330  steps  ? 

17.  A  has  $  150  saved,  and  B  $  100.  How  long  will  it 
be  before  B  has  as  much  saved  as  A,  if  A  is  saving  at  the 
rate  of  $  15  per  month,  and  B  $  17|  per  month  ? 

18.  A  dog  is  100  ft.  behind  a  fox.  The  fox  takes  3 
leaps  while  the  dog  takes  2,  but  2  of  the  dog's  leaps  are 
equal  to  4  of  the  fox's.  How  far  must  the  dog  run  before 
catching  the  fox,  if  the  fox  takes  4  ft.  at  each  leap  ? 

How  far  does  the  fox  go  in  3  leaps  ?  The  dog  in  2  leaps  ?  How 
many  leaps  must  the  dog  take  to  gain  100  ft.  ?  How  many  feet  must 
he  run  to  overtake  the  fox  ? 

19.  A  dog  is  20  yd.  behind  a  fox.  The  fox  takes  5 
leaps  while  the  dog  takes  4,  but  3  of  the  dog's  leaps  equal 
5  of  the  fox's.  How  many  yards  must  the  dog  run  before 
catching  the  fox,  if  the  fox  takes  1\  yd.  at  each  leap  ? 

20.  If  |  of  B's  money  equals  |  of  A's,  and  both  have 
$  112,  how  much  has  each  ? 


ANALYSIS  AND  MENTAL  ALGEBRA  105 

21.  A  rifle  sending  a  ball  1600  ft.  per  second  is  fired 
1  sec.  before  a  rifle  sending  a  ball  2400  ft.  per  second,  and 
in  the  same  direction.  Assuming  that  they  keep  on  at 
the  same  rate,  how  long  will  it  take  the  second  ball. to 
overtake  the  first  ? 

22.  If  a  boat  which  can  travel  10  mi.  an  hour  in  still 
water  is  going  up  a  stream  flowing  4  mi.  an  hour,  how  fast 
does  it  go  ?     How  fast  can  it  go  down  stream  ? 

23.  A  boat  whose  speed  in  still  water  is  8  mi.  per  hour 
makes  a  trip  up  and  back  between  two  cities  on  a  stream 
which  runs  4  mi.  per  hour.  If  it  takes  40  hr.  to  make  the 
round  trip,  what  is  the  distance  between  the  two  cities  ? 

What  is  the  ratio  of  the  rate  of  the  boat  up  stream  to  its  rate 
down  ?  Of  the  time  up,  to  the  time  down  ?  Then  40  hr.  is  how  many 
times  the  time  it  takes  to  go  down  stream  ?  If  it  goes  down  in  \  of 
40  hr.  at  12  mi.  per  hour,  what  is  the  distance  ? 

24.  A  boat  whose  speed  in  still  water  is  121  mi.  per  hour 
takes  50  hr.  to  make  a  round  trip  between  two  points  on 
a  river  running  3|  mi.  per  hour.  What  is  the  distance  be- 
tween the  two  points  ? 

25.  A  boat  can  travel  down  a  certain  stream  3J  times 
as  fast  as  it  can  up  the  stream.  If  the  stream  runs  4  mi. 
per  hour,  what  is  the  speed  of  the  boat  in  still  water  ? 

26.  Two  men  start  from  the  same  place  and  walk  in 
opposite  directions  until  they  are  35  mi.  apart.  If  -|  of 
the  distance  the  first  walks  equals  27  of  the  distance  the 
other  walks,  how  far  does  each  walk  ? 

27.  A  man  agreed  to  work  a  year  for  $  240  and  a  suit 
of  clothes,  but  left  at  tfre  end  of  10  mo.,  receiving  $195 
and  the  suit  of  clothes.     What  was  the  value  of  the  suit  ? 


106  ANALYSIS  AND  MENTAL  ALGEBRA 

How  much  more  would  he  have  received  had  he  worked  2  mo. 
longer  ?  What  then  were  his  wages  for  1  yr.,  if  they  were  $  45  for 
2  mo.  ?  If  his  annual  wages  were  $270  and  $240  was  cash,  what  was 
the  value  of  the  suit  ? 

28.  A  man  was  hired  for  6  mo.,  for  $150  and  a  suit  of 
clothes.  At  the  end  of  4  mo.  he  left,  receiving  $  92  and 
the  suit.     What  was  the  value  of  the  suit  ? 

29.  A  cistern  is  supplied  by  a  pipe  that  can  fill  it  in 
4  hr. ;  it  has  a  discharge  pipe  that  can  empty  it  in  5  hr. 
If  the  cistern  is  empty  and  both  pipes  are  opened,  how 
long  will  it  take  to  fill  it  ? 

What  part  of  the  cistern  would  the  first  pipe  fill  in  1  hr.  ? 

30.  If  a  cistern  has  a  supply  pipe  that  can  fill  it  in  6  hr., 
and  a  discharge  pipe  that  can  empty  it  in  8  hr.,  how  long 
will  it  take  to  fill  the  cistern  if  both  pipes  are  open  ? 

31.  A  cistern  has  a  supply  pipe  that  can  fill  it  in  3  J  hr., 
and  a  discharge  pipe  that  can  empty  it  in  3  hr.  The 
supply  pipe  is  opened,  and  after  it  has  been  running  2  hr., 
the  discharge  pipe  is  opened.  How  long  after  the  latter  is 
opened  will  it  be  before  the  cistern  is  empty  ? 

32.  Two  men  started  from  the  same  place  and  walked 
in  the  same  direction,  and  at  the  end  of  a  certain  time 
were  39  mi.  apart.  If  $  of  the  distance  the  first  walked 
equals  f  of  the  distance  the  second  walked,  how  far  did 
each  walk  ? 

33.  John  can  do  a  certain  piece  of  work  in  8  days; 
with  the  help  of  James  in  5  days.  How  long  should  it 
take  James  alone  to  do  it  ? 

What  is  the  least  number  that  contains  8  and  5  ?  In  40  days  how 
many  times  could  both  do  the  work?    John?    James?    If  James 


ANALYSIS  AND  MENTAL  ALGEBRA  107 

can  do  the  work  3  times  in  40  da.,  in  how  many  days  can  he  do  it 
once? 

Or,  what  part  of  the  work  can  John  do  in  1  day  ?  What  part  can 
John  and  James  together  do  in  1  day  ?  What  part  can  James  alone 
do  in  1  day  ?  How  long,  then,  would  it  take  James  to  do  the  work 
alone  ? 

34.  A  and  B  can  do  a  piece  of  work  in  4  days,  A  alone 
in  7  days.  How  long  should  it  take  B  alone  to  do.  the 
work  ? 

35.  If  John  can  saw  a  cord  of  wood  in  8  hr.,  and  James 
in  10  hr.,  how  long  will  it  take  both  to  saw  a  cord  ? 

How  many  cords  can  John  saw  in  40  hr. ?  James?  Both?  How 
long,  then,  will  it  take  both  to  saw  1  cord? 

Or,  what  part  of  a  cord  can  James  saw  in  1  hr.  ?  John  ?  Both  ? 
How  long,  then,  will  it  take  both  to  saw  1  cord? 

36.  A  can  mow  3  acres  in  2  days,  B  in  3  days.  In  how 
many  days  can  both  together  mow  3  acres  ? 

37.  If  John  can  mow  a  certain  lawn  in  3  hr.,  James  in 
4  hr.,  and  Henry  in  5  hr.,  how  many  hours  will  it  take  all 
three  working  together  ? 

38.  John,  James,  and  Henry  working  together  mow  a 
certain  lawn  in  3  hr.  It  takes  John  and  Henry  5  hr.  to 
mow  the  same  lawn.  In  what  time  could  James  do  it 
alone  ?     (See  problem  33.) 

The  difference  between  the  part  all  do  in  1  hr.  and  the  part  John 
and  Henry  do  in  1  hr.  will  be  the  part  James  can  do  in  1  hr.  From 
this,  how  can  you  find  how  long  it  will  take  James  alone  to  mow  the 
lawn? 

39.  A,  B,  and  C  together  can  shovel  a  carload  of  coal 
in  3  hr.,  and  A  and  B  in  4  hr.  In  how  many  hours  should 
C  be  able  to  do  it  alone  ? 


108  ANALYSIS  AND  MENTAL  ALGEBRA 

40.  A  man  agrees  to  work  for  $2.50  a  day  and  to  for- 
feit $  .50  for  each  working  day  he  is  idle.  If  at  the  end 
of  26  working  days  he  receives  $  47,  how  many  days  has 
he  worked  ? 

How  much  would  he  have  received  had  he  worked  26  days?  How 
much  did  he  lose,  then,  by  idleness?  How  much  did  he  lose  by  each 
day's  idleness  ?  If  he  lost  $  18  by  idleness  and  $  3  each  day  he  was 
idler  how  many  days  was  he  idle?     How  many  did  he  work? 

41.  At  the  end  of  50  days  a  man  received  $  135.  If  he 
had  agreed  to  work  for  $3.50  a  day,  and  to  forfeit  $.50 
each  day  he  was  idle,  how  many  days  was  he  idle  ? 

42.  If  a  man's  expenses  are  $  7  a  week,  and  his  wages 
$  4  a  day,  and  if  at  the  end  of  30  days  he  has  saved  $  70, 
how  many  days  did  he  work? 

43.  What  time  is  it  when  |  of  the  time  past  noon  equals 
|  of  the  time  to  midnight  ? 

44.  Two  men  are  140  rd.  apart.  They  walk  toward 
each  other  till  they  meet,  one  at  the  rate  of  15  rd.  a 
minute,  and  the  other  at  the  rate  of  20  rd.  a  minute. 
How  long  is  it  before  they  meet,  and  how  many  rods  does 
each  walk  ? 

45.  A  and  B  are  40  rd.  apart.  Both  walk  in  the  same 
direction,  reaching  a  certain  point  at  the  same  time.  If  A 
walks  at  the  rate  of  18  rd.  per  minute,  and  B  at  the  rate 
of  22  rd.  per  minute,  how  far  did  each  walk?  How  many 
minutes  did  each  walk  ? 

46.  A  man  started  from  home  on  foot  and  walked  a  cer- 
tain distance  at  the  rate  of  4  mi.  per  hour ;  after  resting 
an  hour  he  rode  back  on  a  street  car  at  the  rate  of  8  mi. 
per  hour.  If  he  was  gone  from  home  7  hr.,  how  far  did 
he  walk?     (See  problem  14.) 


ANALYSIS  AND  MENTAL  ALGEBRA  109 

47.  If  a  hound  is  30  yd.  behind  a  fox,  and  the  fox  takes 
4  leaps  to  the  hound's  3,  2  of  the  hound's  leaps,  of  2  yd. 
each,  being  equal  to  3  of  the  fox's,  how  far  will  the  fox 
run  before  he  is  caught  ? 

48.  If  A,  B,  and  C  together  can  do  a  certain  piece  of 
work  in  3  days,  and  A  and  C  in  5  days,  how  long  would  it 
take  B  alone  to  do  the  work  ? 

49.  A,  B,  and  C  can  build  100  rd.  of  fence  in  4  da.,  A 
and  B  100  rd.  in  6  da.,  and  A  and  C  100  rd.  in  5  da. 
How  many  days  would  it  take  each  one  alone  to  build  100 
rd.? 

50.  If  A  can  do  a  certain  piece  of  work  in  8  da.,  and  A 
and  B  the  same  work  in  5  da.,  how  long  will  it  take  B  to 
finish  the  work  after  A  has  done  f  of  it  ? 

51.  James  can  pile  a  cord  of  wood  in  ^  of  a  day,  and 
Henry  in  J  of  a  day.  How  long  would  it  take  bdth 
together  to  pile  a  cord  ? 

How  many  times  in  1  da.  could  James  pile  a  cord  ?  Henry  ?  Both  ? 
What  part  of  a  day,  then,  will  it  take  both  to  pile  a  cord? 

52.  It  takes  A  \  of  a  day  to  mow  \  A.,  B  \  of  a  day, 
and  C  \  of  a  day.  In  what  time  should  they  be  able  to 
mow  \  A.  together  ? 

53.  If  Richard  is  15  years  old  and  John  8,  how  long 
will  it  be  before  John  is  f  as  old  as  Richard  ? 

54.  How  long  has  it  been  since  a  man  who  is  now  38 
was  20  times  as  old  as  a  boy  who  is  now  19  ? 

55.  A  boy  bought  papers  at  the  rate  of  2  for  3^,  and  as 
many  more  at  the  rate  of  3  for  4^.  He  sold  them  all  at 
the  rate  of  3  for  5^,  thereby  gaining  10  $.  How  many  pa- 
pers of  each  kind  did  he  sell  ? 


110  ANALYSIS  AND  MENTAL  ALGEBRA 

56.  A  grocer  mixes  vinegar  worth  25^  a  gallon  with 
an  equal  amount  of  vinegar  worth  35^,  and,  by  selling 
the  whole  at  40^  a  gallon,  makes  $2.00.  How  many 
gallons  of  each  kind  were  there  ? 

57.  A  and  B  gain  $  550,  of  which  A  receives  $  330.  If 
A's  investment  is  $  300  more  than  ^  of  the  whole  invest- 
ment, what  is  the  whole  investment  ?  What  is  the  invest- 
ment of  each  ? 

58.  A  mounted  messenger  is  sent,  traveling  at  the  rate 
of  9  mi.  per  hour.  Three  hours  later  a  second  one  is  sent 
after  him,  traveling  11  mi.  per  hour.  How  far  must  the 
second  travel  before  overtaking  the  first  ? 

59.  John,  James,  Henry,  and  Richard,  working  sepa- 
rately, can  do  a  certain  piece  of  work  in  1,  1J,  2,  and 
1\  hr.  respectively.  How  long  ought  it  to  take  all  four 
together  ? 

60.  A  man  bought  a  live  turkey  for  %  1.12.  After 
dressing,  it  weighed  fy  as  much  as  when  alive.  He  sold  it 
at  12^  a  pound,  and  gained  13^.  What  was  the  live 
weight  of  the  turkey  ? 

61.  A  and  B  have  $  40,  and  B  has  3  times  as  much  as 
A.     How  much  has  each  ? 

Let  x  represent  the  number  of  dollars  A  has ;  then  3  x  =  the  num- 
ber of  dollars  B  has.  Since  both  have  $  40,  4  x  =  40.  Then  x  =  10, 
the  number  of  dollars  A  has,  and  3  x  =  30,  the  number  B  has. 

Note.  —  x  should  always  represent  a  number. 

62.  A,  B,  and  C  have  together  $  42.  B  has  twice  as 
much  as  A,  and  C  has  3  times  as  much  as  A.  How  much 
has  each  ? 

What  may  x  represent?    How  many  x,  then,  =  42 ? 


ANALYSIS  AND  MENTAL  ALGEBRA  111 

63.  A  line  24  in.  long  is  divided  into  two  parts,  one  of 
which  is  twice  the  other.     How  long  is  each  part  ? 

64.  The  sum  of  the  ages  of  A  and  B  is  55  years,  and  A 
is  15  years  older  than  B.     How  old  is  each  ? 

If  x  years  represents  B's  age,  then  what  represents  A's  age  ? 

Then  x  +  x  +  15  =  what  number? 

If  2  x  +  15  =  55,  what  must  2  x  equal?     xt    x  +  15? 

65.  Divide  60  into  two  parts  that  have  the  ratio  of  2 
to  3. 

If  2  a;  represents  the  smaller  part,  what  represents  the  larger  ? 

66.  How  can  $  28  be  divided  between  A  and  B  so  that 
A  may  have  $  8  more  than  B  ? 

67.  Divide  $  56  among  A,  B,  and  C,  so  that  B  may  have 
1 10  more  than  A,  and  C$6  more  than  B. 

Represent  the  numbers  by  x,  x  +  10,  and  x  +  16. 

68.  Find  two  numbers  whose  difference  is  9,  one  of 
which  is  four  times  the  other. 

If  x  represents  one  number,  what  will  represent  the  other? 

How  will  the  difference  be  represented ?    liix  —  x  =  9,  x=l 

4:X   =    1 

69.  Three  men  in  partnership  gain  $  900.  A's  share  is 
three  times  B's,  and  B's  share  is  twice  C's.  What  is  each 
one's  share  ? 

70.  Three  men  invested  $1200.  A  put  in  twice  as 
much  as  B,  and  C  as  much  as  A  and  B  together.  How 
much  did  each  invest  ? 

71.  A  horse,  carriage,  and  harness  are  together  worth 
$325.  The  horse  is  worth  5  times  as  much  as  the  har- 
ness, and  the  carriage  is  worth  $  45  more  than  the  harness. 
How  much  is  each  worth  ? 


112  ANALYSIS  AND  MENTAL  ALGEBRA 

72.  A  man  being  asked  how  many  sheep  he  had,  replied, 
"  If  I  had  3  times  as  many  as  I  have  and  10  more,  I  should 
have  70."     How  many  had  he  ? 

73.  A  dealer  bought  100  bu.  of  grain.  He  bought  twice 
as  much  wheat  as  oats,  and  2^-  times  as  much  corn  as  wheat 
and  oats  together.  How  many  bushels  of  each  did  he 
buy? 

74.  Divide  69  into  five  parts  in  the  ratio  of  1,  3,  4,  7, 
and  8. 

75.  A  has  $  60,  and  B  has  $  25.  How  much  must  A 
give  B  in  order  that  B  may  have  $  5  more  than  A  ? 

Let  x  =  no.  dollars  A  must  give  B. 

76.  A  has  $  5  more  than  B,  B  has  $  10  more  than  C,  and 
C  has  $  15  more  than  D.  If  all  together  have  $  170,  how 
much  has  each  ? 

Solve  first  by  letting  x  =  the  number  of  dollars  in  D's  share ;  then 
letting  x  =  the  number  of  dollars  in  A's  share.     Which  is  easier  ? 

77.  A  man  walked  10  mi.,  then  drove  a  certain  distance, 
and  then  went  by  train  twice  as  far  as  he  had  driven.  If 
the  whole  distance  traveled  was  70  mi.,  how  many  miles 
did  he  drive  ? 

78.  The  sum  of  two  numbers  is  40  ;  their  difference  is 
12.     What  are  the  numbers  ? 

Let  x  =  one  number ;  then  the  other  =  what  ?    If  x  +  x  +  12  =  40, 

x  =  t 

79.  Divide  100  into  two  parts  such  that  3  times  one 
equals  twice  the  other. 

80.  If  silk  costs  3  times  as  much  as  linen  and  $45  is 
spent  in  buying  20  yd.  of  silk  and  30  yd.  of  linen,  how 
much  does  each  cost  per  yard  ? 


ANALYSIS  AND  MENTAL  ALGEBRA  113 

81.  A  and  B  start  from  the  same  place  and  travel  in  op- 
posite directions  until  they  are  45  mi.  apart.  If  A  travels 
5  mi.  farther  than  B,  how  far  does  each  travel  ? 

82.  A  and  B  start  from  the  same  place  and  travel  in 
opposite  directions  until  they  are  37  mi.  apart.  If  A  travels 
3  miles  less  than  B,  how  far  does  each  travel  ? 

83.  Two  men  have  together  $250.  If  one  has  $30 
more  than  the  other,  how  much  has  each  ? 

84.  If  the  sum  of  two  numbers  is  105  and  their  differ- 
ence is  45,  what  are  the  numbers  ?  What  are  the  num- 
bers, if  the  sum  is  98,  and  the  difference  is  12  ? 

85.  A  and  B  have  together  $  500.  If  A  has  $  120  more 
than  B,  how  much  has  each  ?  How  much  has  each  if  A 
has  $  60  less  than  B  ? 

86.  James  is  4  times  as  old  as  Henry,  and  the  sum  of 
their  ages  is  30  years.     What  is  the  age  of  each  ? 

87.  John  is  4  times  as  old  as  James,  who  is  3  times  as 
old  as  Richard.  If  the  sum  of  their  ages  is  32  years,  how 
old  is  each  ? 

88.  A  father  is  3  times  as  old  as  his  son  ;  in  12  years 
he  will  be  only  twice  as  old.     Find  their  ages. 

If  x  years  is  the  son's  age,  what  is  the  father's  age?  What  is  the 
age  of  each  12  years  hence  ?  What  is  twice  the  son's  age  in  12  years  ? 
If  3a:  +  12  =  2n:  +  24,a:  =  ? 

89.  A's  age  is  6  times  B's,  and  15  years  hence  A  will 
be  only  3  times  as  old  as  B.     What  is  the  age  of  each  ? 

90.  How  can  $  40  be  divided  among  A,  B,  C,  and  D  so 
that  C  may  have  $  2  more  than  D,  B  $  3  more  than  C,  and 
A  $4  more  than  B? 

MCN.  MENT.  AR.  8 


114  ANALYSIS  AND  MENTAL  ALGEBRA 

91.  How  long  is  it  since  a  man  who  is  now  40  years 
old  was  5  times  as  old  as  a  boy  who  is  now  20  years  old  ? 

40  -  x  =  5(20  -  x). 

92.  The  sum  of  two  numbers  is  28,  and  one  of  them 
exceeds  twice  the  other  by  4.     What  are  the  numbers  ? 

93.  If  two  men,  150  miles  apart,  travel  toward  each 
other,  one  at  the  rate  of  8  mi.  per  hour,  and  the  other  at 
the  rate  of  7  mi.  per  hour,  in  how  many  hours  will  they 
meet  ? 

Let  x  represent  the  number  of  hours  each  travels. 

94.  What  number  is  it,  to  which  if  30  is  added,  the  sum 
will  be  4  times  the  original  number  ? 

95.  Five  boys  were  given  52  marbles  so  divided  that 
the  first  received  twice  as  many  as  the  second,  the  second 
3  times  as  many  as  the  third,  the  fourth  twice  as  many  as 
the  first,  and  the  fifth  1^  times  as  many  as  the  second. 
What  was  the  share  of  each  ? 

Why  is  it  best  to  let  x  equal  the  third  number  ? 

96.  A  has  $  25  more  than  B  and  6  times  as  much  as  B. 
How  much  has  each  ? 

97.  The  expenses  of  a  manufacturer  for  3  years  were 
$  21,000.  If  they  increased  $  1000  annually,  what  were 
his  expenses  each  of  the  3  years  ? 

98.  Five  persons  hire  a  coach  for  a  certain  sum.  Had 
there  been  3  more,  the  expense  of  each  would  have  been 
$  1.50  less.     How  much  was  paid  for  the  coach  ? 

If  x  dollars  was  paid  by  each  person,  what  was  paid  by  the  5  persons  ? 
If  each  had  paid  (x  —  1|)  dollars,  how  much  would  the  8  have  paid? 
If  8  a; -12  =  5*,*  =  ? 


ANALYSIS  AND  MENTAL  ALGEBRA  115 

99.  Seven  men  hire  a  coach  for  a  certain  sum.  Had 
there  been  3  more,  the  expense  of  each  would  have  been 
$  3  less.     How  much  was  paid  for  the  coach  ? 

100.  Five  men  go  into  business  together  making  equal 
investments.  Had  there  been  4  more  men,  and  the  total 
investment  the  same,  the  investment  of  each  would  have 
been  $400  less.  What  was  the  investment  of  each  and 
the  whole  investment  ? 

101.  Six  men  lifted  a  steel  rail  of  a  certain  weight.  If 
there  had  been  4  more  men,  each  one  would  have  had  to 
lift  100  lb.  less.     What  was  the  weight  of  the  rail  ? 

102.  Seven  boys  found  a  bag  of  marbles  which  they 
divided  equally.  Had  there  been  only  4  boys,  each  would 
have  had  9  marbles  more.  How  many  marbles  were  there 
in  the  bag  ? 

103.  If  John  is  21  years  old  and  James  8  years,  how 
long  will  it  be  before  John  is  just  twice  as  old  as  James  ? 

How  can  the  age  of  each  at  that  time  (x  years  hence)  be  represented? 

104.  How  many  years  is  it  since  a  man  who  is  now  36 
years  old  was  6  times  as  old  as  a  boy  who  is  16  years  old  ? 

105.  John  is  4  times  as  old  as  James,  and  the  sum  of 
their  ages  is  30  years.  How  long  will  it  be  before  John 
is  only  twice  as  old  as  James  ? 

106.  A's  age  is  twice  B's,  and  B's  is  3  times  C's.  The 
sum  of  their  ages  is  140  years.     What  is  the  age  of  each  ? 

107.  What  time  is  it  when  \  of  the  time  to  noon  equals 
I  of  the  time  past  midnight  ? 

If  x  hours  is  the  time  to  noon,  what  will  represent  the  time  past 
midnight?    If  \x  =  \  of  (12  -  x),  and  2  x  =  12  -  x,  x  =  ? 
Solve  again,  representing  the  time  to  noon  by  4  x  hours. 


116  ANALYSIS  AND  MENTAL  ALGEBRA 

108.  What  time  is  it  when  §  of  the  time  to  noon  equals 
|  of  the  time  past  midnight  ? 

Represent  the  time  past  midnight  by  9  x  hours. 

109.  A  has  $  8  more  than  B ;  and  if  A  had  $  2  more,  he 
would  have  3  times  as  much  as  B.      How  much  has  each  ? 

110.  Two  men  have  together  $  24,  and  the  difference 
between  A's  money  and  B's  equals  half  the  sum.  How 
much  has  each  ? 

ill.  A  and  B  have  together  $  38,  and  A  has  $  2  more 
than  twice  as  much  as  B.     How  much  has  each? 

112.  A  man  divided  his  property  among  his  three  chil- 
dren so  that  the  eldest  received  twice  as  much  as  the  sec- 
ond, and  the  second  3  times  as  much  as  the  youngest.  If 
the  eldest  received  $  5000  more  than  the  youngest,  how 
much  did  each  receive,  and  what  was  the  whole  amount 
divided  ? 

113.  Divide  $  200  among  5  men,  4  women,  and  3  chil- 
dren, so  that  each  woman  receives  $  10  more  than  each 
child,  and  each  man  $  10  more  than  each  woman. 

114.  In  a  mixture  of  86  gal.  of  wine  and  water,  there 
were  18  gal.  more  water  than  wine.  How  much  was  there 
of  each  ? 

115.  A  dealer  bought  810  bu.  of  grain.  He  bought  3 
times  as  much  wheat  as  oats,  and  1J  times  as  much  corn 
as  wheat  and  oats  together.  How  much  of  each  did  he 
buy? 

116.  Five  men  have  together  $  12,000,  and  their  shares 
are  in  the  ratio  1,  1^,  2,  2J,  and  5.  What  is  the  share  of 
each? 


ANALYSIS  AND  MENTAL  ALGEBBA  117 

117.  Find  two  numbers  whose  difference  is  25,  one  of 
which  is  6  times  the  other. 

118.  A  horse,  carriage,  and  harness  are  together  worth 
$800.  The  carriage  is  worth  $100  more  than  the  har- 
ness, and  the  horse  $  100  less  than  twice  as  much  as  the 
carriage  and  harness  together.     How  much  is  each  worth  ? 

119.  Four  men  bought  a  piece  of  property,  investing  equal 
amounts.  If  3  men  had  bought  the  property,  each  would 
have  had  to  invest  $  200  more.  How  much  was  the  prop- 
erty worth  ? 

•  120.   What  time  is  it  when  -|  of  the  time  past  noon 
equals  $  of  the  time  to  midnight  ?     (See  problem  108.) 

121.  At  what  time  between  3  and  4  o'clock  are  the 
hour  and  minute  hands  of  a  clock  exactly  together  ? 

a.  How  many  minute  spaces  does  the  minute  hand  travel"  while 
the  hour  hand  goes  one  ?  Then,  in  moving  12  minute  spaces  while 
the  hour  hand  moves  one,  how  many  minute  spaces  does  the  minute 
hand  gain  on  the  hour  hand  ?  If  it  has  to  move  12  spaces  to  gain  11, 
how  many  spaces  must  it  move  to  gain  one  ?  At  3  o'clock  how  many 
minute  spaces  has  it  to  gain  ?  If  it  has  to  move  1^  spaces  to  gain 
one,  how  many  must  it  move  to  gain  15  ?  What  time  is  it,  then,  when 
the  minute  hand  has  moved  16^  minute  spaces  from  its  position  at  3 
o'clock? 

b.  Think  of  the  position  of  the  hands  at  3  o'clock.  Let  x  represent 
the  number  of  minute  spaces  the  hour  hand  has  to  move.  Then,  since 
the  minute  hand  moves  12  times  as  fast,  what  will  represent  the  num- 
ber of  minute  spaces  the  minute  hand  has  to  move  ?  Since  they  are 
15  spaces  apart  at  3  o'clock,  what  must  be  added  to  x  to  make  it  equal 
12  *?    H  12a;  =  :c+15,  x  =  ? 

Which  solution  is  the  easier  ? 

122.  At  what  time  between  4  and  5  o'clock  are  the  hour 
and  minute  hands  together  ?     (Solve  in  two  ways.) 


118  ANALYSIS  AND  MENTAL  ALGEBBA 

123.  At  what  time  between  2  and  3  o'clock  are  the  hour 
and  minute  hands  together  ?  Between  1  and  2  o'clock  ? 
Between  8  and  9  o'clock  ?     Between  11  and  12  o'clock  ? 

124.  A  and  B  invest  a  certain  amount  in  business  and 
gain  $  500.  A's  investment  is  f  of  the  whole  lacking  $  90, 
and  his  share  of  the  gain  is  $  300.  What  is  the  whole 
investment,  and  the  investment  of  each  ? 

a.  What  part  of  the  gain  is  A's?  Then,  what  part  of  the  whole 
investment  is  A's?  If  A's  investment  is  f  of  the  whole,  and  also 
|  of  the  whole  lacking  $  90,  the  $  90  is  what  part  of  the  whole  invest- 
ment ?    If  $  90  is  ^  of  the  whole,  what  is  the  whole  ? 

b.  If  x  dollars  is  the  whole  investment,  f  of  x  —  90  =  f  of  re. 
Complete  the  solution. 

125.  A  and  B  gain  $450,  of  which  A  receives  $200. 
If  A's  investment  is  §  of  the  whole  lacking  $  200,  what  is 
the  whole  investment  and  the  investment  of  each  ? 

126.  Two  men  cut  a  certain  amount  of  wood  for  $  60. 
The  first  received  $  25,  and  he  cut  J  of  the  whole  lacking 
3  cd.  What  was  the  whole  amount  cut,  and  the  amount 
each  cut  ? 

127.  Two  men  cut  a  certain  amount  of  wood  for  $  60. 
The  second  received  $  35,  and  he  cut  J  of  the  whole  and 
3  cd.  more.  What  was  the  whole  amount  cut,  and  the 
amount  each  cut  ? 

128.  Seven  persons  hire  a  coach  for  a  certain  sum. 
Had  there  been  3  more,  the  expense  of  each  would  have 
been  $  3  less.     What  was  paid  for  the  coach  ? 

a.  Had  3  more  gone,  how  much  would  the  expense  of  the  7  persons 
have  been  lessened  ?  The  3  would  have  paid  how  much  then  ?  The 
10  how  much  ? 

b.  See  problem  98. 


ANALYSIS  AND  MENTAL  ALGEBRA  119 

129.  A  sum  of  money  was  found  by  5  persons  and 
divided  equally.  Had  there  been  10  more  persons,  each 
would  have  received  $  10  less.  How  much  money  was 
found  ? 

130.  A  sum  of  money  was  found  by  7  persons.  Had 
there  been  only  4  persons,  each  would  have  had  $  6  more. 
What  was  the  sum  ? 

131.  A  man  agreed  to  work  a  year  for  $360  and  a  suit 
of  clothes.  He  left  at  the  end  of  9  mo.,  receiving  $261 
and  the  suit.     What  was  the  value  of  the  suit  ? 

a.  See  problem  27. 

b.  Represent  the  value  of  the  suit  by  x  dollars. 

132.  A  grocer  mixed  tea  that  cost  48/  a  pound  with 
an  equal  amount  of  50  ^  tea,  and  sold  the  mixture  at  55  $ 
a  pound,  thereby  gaining  $4.80.  How  many  pounds  of 
each  kind  were  there  ?     (Solve  by  analysis.) 

133.  A  boy  bought  oranges  at  the  rate  of  4  for  5  ^,  and 
as  many  more  at  the  rate  of  5  for  6  ^.  He  sold  them  all 
at  the  rate  of  2  for  3  ^,  thereby  gaining .  11  ^.  How 
many  did  he  buy  ?     (Solve  by  analysis.) 

134.  A  and  B  had  the  same  amount  of  land.  A  bought 
20  A.  from  B  and  sold  10  A.  to  B,  and  then  had  3  times 
as  much  as  B.     How  much  had  each  at  first  ? 

135.  A  and  B  are  24  rd.  apart,  and  are  walking  toward 
each  other.  A  takes  3  steps  while  B  takes  4,  and  3  of  A's 
steps  equal  2  of  B's.  How  far  will  each  walk  before  they 
meet?     (Solve  by  analysis.) 

136.  A  is  22  yr.  older  than  B,  and  f  of  A's  age  equals 
|  of  B's.     How  old  is  each  ? 


120  ANALYSIS  AND  MENTAL  ALGEBRA 

137.  Two  men  start  from  the  same  place  and  walk  in 
the  same  direction,  and  after  a  certain  time  are  10  mi. 
apart.  How  far  does  each  walk  if  |  of  the  first  one's 
distance  equals  -|  of  the  second  one's  distance  ? 

138.  If  |  of  A's  money  equals  f  of  B's,  and  both  have 
$  147,  how  much  has  each  ? 

a.   See  problem  1.     b.  Solve,  using  x. 

139.  A  and  B  have  together  $  68 ;  |  of  A's  money  equals 
|  of  B's.     How  much  has  each  ? 

140.  A  and  B  have  together  $95,  and  A  has  4  times 
as  much  as  B.     How  much  has  each  ? 

141.  A,  B,  and  C  have  $120.  A  has  twice  as  much  as 
C  and  $5  more ;  B  has  3  times  as  much  as  C,  lacking  $  5. 
How  much  has  each  ?     (Solve  by  analysis.) 

142.  A,  B,  and  C  can  do  a  certain  piece  of  work  in  20 
days ;  A  and  B  in  30  days ;  B  and  C  in  40  days.  How 
long  will  it  take  each  to  do  the  work  alone  ? 

143.  A  line  90  in.  long  is  divided  into  2  parts,  one  of 
which  is  f  of  the  other.     How  long  is  each  ? 

144.  A  and  B  have  $520,  and  A  has  $75  more  than  B. 
How  much  has  each  ? 

145.  If  a  man  has  property  to  the  amount  of  $36,000, 
and  debts  to  the  amount  of  $16,000,  how  much  is  he 
worth?  How  much  is  he  worth  if  his  debts  are  $36,000 
and  his  property  $16,000  ? 

146.  If  James  had  4  times  as  many  marbles  and  10  more, 
he  would  have  102.     How  many  has  he  ? 

147.  The  sum  of  two  numbers  is  80 ;  their  difference  is 
38.     What  are  the  numbers  ? 


ANALYSIS  AND  MENTAL  ALGEBRA  121 

148.  John  has  $500,  and  James  $600.  How  long  will 
it  be  before  John  has  $100  more  than  James,  if  he  saves 
at  the  rate  of  $18  a  month,  and  James  at  the  rate  of  $13  a 
month  ? 

149.  A  cistern  has  a  supply  pipe  that  can  fill  it  in  5  hr., 
and  a  discharge  pipe  that  can  empty  it  in  3J  hr:  If  the 
cistern  is  full  and  both  pipes  are  opened,  how  long  will  it 
take  to  empty  the  cistern  ? 

150.  At  what  time  between  7  and  8  o'clock  are  the 
hands  of  a  clock  together  ? 

151.  A  and  B  had  the  same  amount  of  money.  A  gave 
B  $10,  and  then  B  had  3  times  as  much  as  A.  How  much 
had  each  at  first  ? 

152.  A  man  agreed  to  work  for  $3  a  day  and  to  forfeit 
60  ^  for  each  working  day  he  was  idle.  At  the  end  of  45 
working  days  he  received  $81.  How  many  days  did  he 
work? 


SECTION   XIII 

REVIEWING  AND  EXTENDING  THE   IDEA  OF 
PERCENTAGE 

SUGGESTIONS 

a.  In  the  study  of  the  problems  of  this  section  the  learner  should, 
in  all  cases  of  difficulty,  go  back  to  preceding  sections  to  review  the 
foundation  of  each  idea  which  is  not  clear.  After  following  the  devel- 
opment of  each  fundamental  idea  in  a  given  problem,  the  pupil  will 
find  it  of  great  value  in  preparing  for  the  recitation  to  diagram  the 
problem  so  that  there  will  be  a  clear  image  in  the  mind  when  oral 
analysis  is  attempted. 

b.  In  the  study  of  the  lesson  before  the  class  is  called,  the  learner 
will  get  much  help  from  thinking  the  problem  through  in  words  as 
well  as  in  images.  Words  embodied  in  short,  clear  sentences  help 
much  in  making  plain  to  other  members  of  the  class  the  analyses 
presented. 

c.  In  the  explanation  of  difficult  problems,  if  the  word  used  does 
not  call  up  clear  images  in  the  minds  of  other  members  of  the  class, 
the  pupil  who  gives  the  analysis  should  be  afforded  an  opportunity  to 
place  on  the  board  for  the  benefit  of  others  the  pictures  or  diagrams 
he  knows  to  be  pertinent  to  the  expression  of  the  thoughts  in  his 
mind. 

d.  At  the  end  of  this  section,  special  applications  of  percentage  to 
insurance,  to  stocks  and  bonds,  and  to  discounts  have  been  placed  for 
the  purpose  of  leading  pupils  to  see  that  all  operations  in  this  division 
can  be  based  on  the  fundamental  notions  already  established. 

l.  What  per  cent  of  the  cost  of  -|  of  a  quantity,  is  the 
cost  of  |-  of  the  quantity  ? 

122 


EXTENDING   THE  IDEA   OF  PERCENTAGE         123 

2.  A  man  owning  f  of  a  mill  sold  f  of  what  he  owned. 
What  per  cent  of  the  mill  did  he  still  own  ? 

3.  John's  marbles  are  62|%  of  James's.  If  James 
has'  32,  how  many  has  John  ?  If  John  has  25,  how  many 
has  James  ? 

4.  A,  B,  C,  and  D  engage  in  business.  A  invests 
$500,  B  11000,  C  $1500,  and  D  $2000.  At  the  end  of  a 
year  they  have  gained  $2000.  What  per  cent  of  the 
gain  should  each  one  have? 

5.  18f  %  of  a  certain  sum  is  $18.  What  is  371%  of 
it?  33i%?  75%?  83i%?  41|%?  43f  %  ?  68f %  ? 
50%? 

6.  One  number  is  81^%  of  another.  If  the  larger 
number  is  80,  what  is  the  other?  If  the  smaller  is  78, 
what  is  the  larger  ? 

7.  A,  B,  and  C  engaged  in  business.  A  furnished 
331%  of  the  capital,  and  B  43|%  of  it.  What  part  did  C 
furnish  ? 

8.  A  farmer  sold  2  cows  at  $24  each.  On  one  he 
gained  20%,  and  on  the  other  he  lost  20%.  How  much 
did  each  cost  ? 

9.  A  owned  f  of  a  mill  and  sold  75%  of  his  share. 
What  per  cent  of  the  mill  did  he  still  own  ? 

10.  At.  what  price  must  goods  costing  $200  be  sold, 
in  order  to  gain  121%  ?     I6f  %  ?     41|%  ? 

11.  B  has  75  A.  of  land,  which  is  37|  %  of  what  A  has. 
How  much  land  has  A  ? 

12.  An  agent  sells  cotton  for  a  planter,  receiving  2% 
commission.  How  much  should  the  agent  remit  to  the 
owner  if  the  selling  price  was  $  4000  ? 


124  EXTENDING   THE  IDEA   OF  PERCENTAGE 

Commission  is  a  percentage  allowed  an  agent  for  buying  or  selling 
for  another,  and  is  usually  estimated  on  the  amount  of  money  the 
agent  handles. 

13.  An  agent  buys  1000  sacks  of  flour  at  $  2  per  sack. 
What  is  his  commission  at  2J%  ? 

14.  An  agent  received  1 45  for  buying  flour.  If  his 
commission  was  2J%,  what  was  the  value  of  the  flour? 
How  much  did  it  cost  the  owner  (commission  included) 
per  sack,  if  there  were  1000  sacks  ? 

15.  How  much  money  must  a  merchant  send  his  agent 
in  order  that  he  may  buy  $  1000  worth  of  goods,  commis- 
sion 1J%? 

16;  A  man  sent  his  agent  $  1025.  The  agent  invested 
$  1000,  and  retained  the  remainder  as  commission.  What 
was  the  rate  of  commission  ? 

17.  At  6%  interest  per  year,  what  is  the  rate  per  cent 
for  1  yr.  4  mo.?     1  yr.  2  mo.?%    3  yr. ?     4  yr.  6  mo.? 

1  yr.  1  mo.  ? 

Interest  is  money  paid  for  the  use  of  money,  and  is  usually  reckoned 
as  so  many  per  cent  of  the  principal  (the  amount  loaned)  per  annum, 
or  year.  Then  since  1  yr.  4  mo.  =  f  yr.,  the  rate  per  cent  in  the  first 
case  is  $  of  6  %,  or  8  %. 

18.  What  is  the  interest  on  $400  for  1  year  at  6%  per 
annum  ?     For  2  years  at  the  same  rate  ?     1  yr.  6  mo.  ? 

2  yr.  8  mo.  ?     4  yr.  1  mo.  ?     2  yr.  30  da.  (=1  mo.)? 

19.  At  6%  per  annum,  give  the  interest  on  : 
$  100  for  2  yr.  6  mo.  $1000  for  16|  yr. 

1 1000  for  31  yr.  $  32  for  3  yr.  1  mo.  15  da. 

$  240  for  2  yr.  30  da.  $  50  for  3  mo. 

$  300  for  5  yr.  $40  for  7  mo.  15  da. 

$  1500  for  10  yr.  $  375  for  5  mo.  10  da. 


EXTENDING   THE  IDEA   OF  PERCENTAGE         125 

20.  What  is  the  interest  of  a  principal  of  $200,  at  4%, 
for  2  yr.  6  mo.  ? 

21.  If  the  interest  is  120,  time  2 £  yr.,  and  rate  4%, 
what  is  the  principal  ? 

What  is  the  rate  per  cent  at  4  %  per  annum  for  1\  yr.  ?  If  $  20  is 
10%  of  the  principal,  what  is  100%? 

22.  Find  the  principal  when  the  interest  is  $12,  time 

2  yr.  6  mo.,  rate  6%.     When,  the  interest  is  $.75,  time 

3  mo.,  rate  6%. 

23.  A  man  borrowed  a  sum  of  money  for  6  mo.,  and 
when  paying  it  back  at  the  end  of  the  time  he  gave  back 
$6  more  than  he  received.  If  the  rate  was  12%,  how 
much  money  did  he  borrow?  If  the  rate  had  been  6%, 
what  would  the  principal  have  been  under  the  same 
conditions  ? 

24.  On  Sept.  1,  Mr.  B  paid  the  interest  on  two  sums  of 
money  he  had  borrowed,  $  24  in  each  case.  The  first  sum 
had  been  borrowed  1  yr.  4  mo.  before  at  6%,  the  other 
1  yr.  6  mo.  before  at  8%.  What  were  the  two  sums 
borrowed  ? 

25.  What  per  cent  of  a  principal  of  $  250  is  $  25  interest  ? 
If  the  time  was  1  yr.  8  mo.,  what  was  the  rate  per  annum  ? 
If  the  rate  per  annum  was  6%,  what  was  the  time  in 
years  ? 

26.  What  per  cent  of  a  principal  of  $500  is  $110 
interest  ?  If  the  time  was  2  yr.  9  mo.,  what  was  the  rate 
per  annum  ?  If  the  rate  had  been  12%,  what  would  the 
time  have  been  ?  ♦ 


126  EXTENDING   THE  IDEA   OF  PERCENTAGE 

27.  Given  principal  $  200,  time  1  yr.  6  mo.,  interest  $  30  ; 
find  rate.  • 

Given  principal  §200,  time  2  yr.  6  mo.,  interest  $20; 
find  rate. 

Given  principal  $  480,  time  2  yr.  1  mo.,  interest  $60; 
find  rate. 

Given  principal  $  1000,  time  2  yr.,  interest  $  100  ;  find 
rate. 

Given  principal  $  3600,  time  1^  yr.,  interest  $  360  ;  find 
rate. 

28.  Given  principal  $200,  interest  $30,  rate  10%;  find 
time. 

Given  principal  $75,  interest  $15,  rate  8%;  find  time. 
Given  principal  $60,  interest  $20,  rate  8%;  find  time. 
Given  principal  $50,  interest  $.75,  rate  6%;  find  time. 
Given  principal  $  100,  interest  $  5,  rate  2|  % ;  find  time. 

29.  In  what  time  will  $100  produce  $100  interest  at 
6%?  In  what  time  will  any  principal  double  itself  at  6%? 
At  8%?     At  4%?     At  7%? 

30.  A  merchant  wishes  to  buy  goods  to  the  amount  of 
$1000.  He  borrows  enough  money  for  3  mo.  at  8%  to 
include  the  cost  of  the  goods  and  a  commission  of  2J%  to 
his  agent.  At  the  end  of  the  3  mo.  how  much  must  he 
pay  back  to  the  lender  ? 

31.  For  how  much  must  hats  costing  $  30  per  dozen  be 
sold  in  order  to  gain  20%? 

32.  A  boy  buys  papers  at  the  rate  of  2  for  3  cents,  and 
sells  them  for  2  cents  apiece.  What  per  cent  does  he 
gain  ? 


EXTENDING   THE  IDEA   OF  PERCENTAGE         127 

33.  A  boy  sells  papers  for  3  ^  each,  thereby  gaining  50%. 
How  many  can  he  buy  for  6  ^  ? 

34.  In  order  to  gain  12-|-%,  hats  are  marked  at  $2.70. 
What  is  the  cost  ? 

35.  A  dealer's  price  for  a  carriage  was  $128,  but  in 
order  to  make  a  sale  he  made  a  discount  of  121%.  How 
much  did  the  carriage  cost  if  he  still  gained  16J  %?  How 
much  would  he  have  gained  had  he  sold  it  at  the  price  he 
asked  ? 

36.  48  is  8  times  18f%  of  what  number? 

37.  What  per  cent  of  1  bu.  3  pk.  is  3  pk.  4  qt.  ? 

38.  41 J  %  of  one  number  equals  31^%  of  another.  If 
their  sum  is  280,  what  are  the  numbers  ? 

39.  How  much  must  a  bicycle  costing  $  48  be  marked  in 
order  that  a  discount  of  16f  %  may  be  made  from  the 
price  marked,  and  a  gain  of  25%  still  be  made? 

40.  A  man  sold  two  horses  for  $126  each.  On  one  he 
gained  121%,  and  on  the  other  he  lost  12|%.  Did  he 
gain  or  lose,  and  how  much  ? 

41.  Which  will  be  greater  and  how  much,  the  interest 
on  $200  for  2  yr.  6  mo.  at  10%,  or  the  interest  on  $300 
for  1  yr.  6  mo.  at  12%? 

42.  What  sum  of  money  was  borrowed,  if  the  interest 
at  9%  for  1  yr.  8  mo.  was  $  75  ? 

43.  What  is  the  ratio  of  the  interest  on  $  500  for  1  yr. 
9  mo.  at  101%,  to  the  interest  on  $1000  for  2  yr.  4  mo. 
at  7%? 

What  is  the  ratio  of  the  first  principal  to  the  second?  Of  the  first 
time  to  the  second?  Of  the  first  rate  to  the  second?  Of  the  first 
interest  to  the  second  ? 


128         EXTENDING  THE  IDEA   OF  PERCENTAGE 

44.  What  per  cent  of  $  is  f  ?  Of  f  is  §?  Of  £  is  J? 
Of  J  is  f?     Off  is*?     Off  is  4? 

45.  An  agent's  commission  for  selling  a  piece  of  real 
estate  was  3%.  If  he  received  8  270,  what  was  the  value 
of  the  property? 

46.  A  dealer's  gain  on  a  horse  was  1 25,  which  was  a 
gain  of  16f%.     What  was  the  cost  and  the  selling  price  ? 

47.  If  a  grocer  sells  f  of  a  dozen  eggs  for  what  -J  of  a 
dozen  cost  him,  what  is  his  gain  per  cent? 

48.  How  much  must  cloth  that  cost  $1.25  per  yard  be 
marked  so  that  a  discount  of  20%  may  be  made,  and  the 
gain  be  60%  ? 

49.  A  man  wishing  to  mortgage  his  farm  for  $1000, 
has  to  pay  the  agent  a  commission  of  2^%  and  the  inter- 
est for  6  mo.  in  advance  at  6% .  How  much  does  he  receive 
from  the  agent? 

50.  What  number  is  it  of  which  62^%  exceeds  41|% 
by  100? 

51.  What  per  cent  of  A's  money  is  B's,  if  B  has  16|% 
more  than  A?     What  per  cent  of  B's  is  A's? 

52.  What  per  cent  of  A's  money  is  B's  if  B  has  16f  % 
less  than  A?     What  per  cent  of  B's  is  A's? 

53.  A  man  had  a  note  due  that  had  run  for  90  da.  at 
6%.  If  he  paid  the  lender  $101.50,  what  was  the  amount 
borrowed? 

54.  A  grocer  sold  sugar  at  9/  a  pound,  and  gained 
121%.     How  much  did  it  cost  him? 

55.  Goods  were  sold  at  auction  at  a  loss  of  37^%  for 
$35.     What  was  the  cost? 


EXTENDING   THE  IDEA   OF  PERCENTAGE         129 

56.  If  |  of  a  lot  of  goods  is  sold  for  what  J  cost,  what 
per  cent  is  gained? 

57.  If  a  cow  is  worth  50%  of  the  value  of  a  horse,  a  hog 
37  J  %  of  the  value  of  a  cow,  and  a  sheep  41f  %  of  the  value 
of  a  hog,  how  much  is  the  horse  worth,  the  sheep  being 
valued  at  $2.50? 

58.  At  what  rate  must  $100  be  loaned  in  order  to  bring 
in  $  7  in  1  yr.  2  mo.  ? 

59.  How  much  interest  must  be  paid  semiannually  on 
a  12000  mortgage  at  7%  ? 

60.  What  is  the  ratio  of  the  interest  of  $  350  for  6  yr. 
2  mo.  at  13%,  to  the  interest  of  $490  for  3  yr.  1  mo. 

at  6|%? 

61.  If  A  has  37|%  more  money  than  B,  what  per  cent 
of  B's  money  is  A's?     What  per  cent  of  A's  is  B's? 

62.  37  J  %  of  A's  money  equals  b<6\%  of  B's.  If  A  has 
$10  more  than  B,  how  much  has  each? 

NOTES  ON  THE  APPLICATIONS  OF  PERCENTAGE   TO 
INSURANCE 

a.  An  agreement  by  one  party  to  protect  another  from  loss  or 
damage  for  a  consideration  is  usually  called  Insurance.  The  contract 
between  the  insurance  company  and  the  party  insured  is  the  Policy. 
The  sum  paid  for  protection  is  the  Premium. 

b.  The  pupil  who  can  keep  the  special  terms  in  mind  and  see  in 
them  new  names  for  old  ideas  will  find  no  difficulty  in  solving  prob- 
lems in  insurance. 

63.  The  face  of  an  insurance  policy  is  $2400;  the 
rate  is  1%.     What  is  the  premium? 

MCN.   MENT.  AR.  —  9 


130  EXTENDING   THE  IDEA   OF  PERCENTAGE 

64.  A  building  is  insured  for  two  thirds  of  its  value : 
the  rate  of  insurance  is  2%  ;  the  value  of  the  building  is 
$1500.  What  is  the  premium?  What  is  the  face  of  the 
policy  ? 

65.  What  will  be  the  premium  for  insuring  $2000 
worth  of  wheat  for  one  half  its  value  at  ^%?  At  f  %? 
What  will  be  the  face  of  the  policy? 

66.  What  will  be  the  premium  for  insuring  f  of  the 
value  of  a  house  worth  $600  at  £%?  What  will  be  the 
face  of  the  policy? 

67.  At  a  rate  of  1%  what  insurance  will  be  written 

for  $75? 

68.  At  a  rate  of  3%  what  insurance  will  be  written 

for  $60? 

69.  At  a  rate  of  4^%  what  insurance  will  be  written 
for  $9? 

70.  At  a  rate  of  |%  what  insurance  .will  be  written 
for  $12? 

71.  At  a  rate  of  1-|%  what  insurance  will  be  written 
for  $36? 

NOTES  ON  THE  APPLICATIONS  OF  PERCENTAGE 
TO  STOCKS  AND  BONDS 

a.  When  sold  at  the  value  written  or  printed  on  the  face,  stocks 
and  bonds  are  rated  at  par.  When  they  are  sold  at  a  price  higher 
than  that  stated  on  the  face,  they  are  rated  above  par.  When  sold  at 
a  price  lower  than  that  stated  on  the  face,  stocks  and  bonds  are  said 
to  be  below  par. 

b.  The  Par  Value  may  be  any  amount  agreed  upon  and  authorized 
by  general  law  or  by  the  Charter  which  defines  the  powers  and  limita- 
tions of  a  company. 


EXTENDING   THE  IDEA   OF  PERCENTAGE         131 

c.  When  the  business  of  a  company  is  conducted  at  a  profit,  the 
gain  is  distributed  among  the  stockholders,  and  each  one's  share  of 
the  gain  is  called  a  Dividend.  When  business  is  done  at  a  loss,  stock- 
holders are  required  to  make  up  the  loss  by  an  Assessment.  Dividends 
and  assessments  are  usually  apportioned  at  some  per  cent  upon  the 
par  value. 

d.  The  money  or  commission  paid  a  broker  for  buying  or  selling 
stocks  and  bonds  is  called  Brokerage.  Brokerage  is  estimated  on  the 
par  value. 

e.  The  learner  has  to  use  new  names  for  ideas  already  in  his  mind. 
The  problems  in  stocks  and  bonds  are  easy  ones  in  simple  percentage. 

72.  How  much  capital  has  a  man  in  a  company  if  he 
holds  500  shares,  having  a  par  value  of  $  10  ? 

73.  What  is  the  market  value  of  400  shares  of  stock 
with  a  par  value  of  $10,  if  the  stock  is  quoted  at  5% 
above  par  (at  105%)  ?     If  quoted  at  5 %  below  par? 

74.  How  much  must  I  pay  for  bonds,  quoted  at  59^, 
if  I  have  to  pay  a  brokerage  of  ^  per  cent,  and  the  par 
value  is  $  100  ? 

75.  If  current  interest  is  4%,  and  Black  Rock  Mining 
Stock  is  paying  25  %  annual  dividends,  is  the  stock  worth 
more  or  less  than  par  ?     Why  ? 

76.  What  assessment  must  a  man  pay  if  he  holds 
10,000  shares  in  a  company  of  $  100,000  capital,  if  the 
company  is  obliged  to  make  an  assessment  to  meet  a  loss 
of  $  20,000,  the  par  value  of  his  stock  being  $  1  ? 

77.  What  dividend  would  a  man  receive  if  he  holds 
10,000  shares,  par  value  being  $  1,  in  a  company  with  a 
capital  of  $50,000,  if  the  net  profits  for  one  year  are 
$10,000? 


132  EXTENDING   THE  IDEA   OF  PERCENTAGE 

78.  A  6  %  dividend  of  $  12,000  was  distributed  among 
the  stockholders  of  a  company.  What  was  the  capital 
stock  of  the  corporation  ? 

79.  A  4  %  assessment  of  $  8000  was  levied  upon  the 
stockholders  of  a  company.  What  was  the  capital  stock 
of  the  company  ? 

80.  If  stock  is  selling  at  25  %  above  par,  and  I  invest 
$  5000  in  it,  what  is  the  par  value  of  my  investment  ? 

81.  If  stock  is  selling  at  20  %  below  par,  and  I  invest 
$  4000  in  it,  what  is  the  par  value  of  my  investment  ? 

NOTES  ON  THE   APPLICATIONS  OF  PERCENTAGE 
TO   DISCOUNTS 

a.  A  deduction  from  the  price  of  an  article,  from  the  amount  of  a 
bill,  or  from  the  face  of  a  note,  is  called  Discount.  The  fixed  price  is 
called  the  List  Price.     The  list  price,  less  the  discount,  is  the  Net  Price. 

The  difference  between  the  face  of  a  bank  note*  and  what  the  bor- 
rower receives  is  the  Bank  Discount.  What  a  borrower  receives  on  his 
note  given  to  a  bank  is  the  Proceeds. 

b.  A  discount  of  15  %  off  means  a  deduction  of  15%  from  the  price 
of  an  article  or  from  the  face  of  a  bill.  A  discount  of  20  %  and  10  % 
does  not  mean  30  %  off,  but  that  20  %  must  first  be  deducted,  leaving 
80  %  of  the  price,  then  10  %  of  80  %  of  the  price,  leaving  72  %  of  the 
price  of  the  article  or  the  face  of  the  bill  to  be  paid.  "  3  tens  and 
2  %  off  "  means  three  successive  discounts  of  10  %,  and  2  %  from  the 
remainder,  etc. 

c.  Bank  Discount  is  interest  taken  in  advance.  The  term  of  discount 
corresponds  to  the  time  in  interest.  The  custom  as  to  days  of  grace 
is  not  uniform  in  all  parts  of  the  United  States.  The  expiration  of 
the  term  of  discount,  including  three  days  of  grace,  where  they  are 
allowed,  is  the  Maturity  of  the  note. 

d.  If  the  pupil  can  interpret  his  old  ideas  in  the  terms  of  discount, 
he  will  have  no  difficulty  in  solving  and  analyzing  the  problems  given. 


EXTENDING   THE  IDEA    OF  PERCENTAGE         133 

82.  What  is  the  net  cost  of  a  bill  of  goods  for  $600, 
bought  at  10%  discount,  and  5%  off  for  cash? 

83.  Find  the  net  price  and  discount  on  a  bill  of  goods 
for  9 400  at  20%  and  10%  off  for  cash? 

84.  Find  what  was  paid  on  a  bill  for  $300  at  331%, 
25%,  and  10%  off.     What  was  the  discount? 

85.  What  is  the  list  price  of  a  book  which  is  sold  for 
two  thirds  of  a  dollar  after  a  discount  of  33  J  %  has  been 
taken  off  ? 

86.  What  is  the  difference  between  a  discount  of  30% 
and  a  discount  of  25%  and  5%  off  on  a  bill  for  $  120  ? 

87.  What  is  the  amount  of  a  bill,  which  after  a  discount 
of  121%,  amounts  to  $  14  ? 

88.  What  are  the  proceeds  of  a  note,  discounted  at  a 
bank  for  60  days,  no  grace,  for  $  80,  at  6  %  ?  What  is  the 
bank  discount  ? 

89.  Find  the  face  of  a  note,  which  after  being  dis- 
counted at  a  bank  for  four  months  at  6%,  no  grace  con- 
sidered, will  give  $  49  proceeds. 

90.  For  what  sum  must  a  note,  due  in  6  months  at  4%, 
be  drawn  to  yield  $  196  proceeds,  no  grace  considered  ? 

91.  A  man  owes  a  bill  for  $  100  which  is  not  due  for  3 
months.  He  is  offered  a  discount  of  3%  off  for  cash.  If 
he  should  borrow. money  at  the  bank  at  8%  to  pay  the 
bill,  would  he  gain  or  lose,  and  how  much,  no  grace  being 
considered  ? 


SECTION   XIV 

REVIEWING    AND    EXTENDING    IDEAS    PREVIOUSLY 
PRESENTED  AND  ESTABLISHING  NEW  RELATIONS 

SUGGESTIONS 

a.  In  advancing  through  this  section,  the  learner  must  see,  in 
his  private  study,  the  fundamental  ideas  in  his  problems,  in  order  to 
give  oral  expression  to  them.  In  many  cases,  the  converse  of  ideas 
established  in  the  preceding  sections  of  the  book  is  given.  Economy 
of  effort  will  follow  if,  through  diagrams  made  during  the  hour  for 
preparation,  the  new  or  changed  relations  are  brought  clearly  before 
the  mind. 

b.  The  learner  in  preparing  the  lesson  should  realize  that  when 
he  knows  relations,  he  can  through  proper  study  express  his  knowledge 
so  as  to  bring  to  the  minds  of  others  his  own  images  and  thoughts. 

c.  In  studying  a  problem,  the  learner  should  always  seek  for  the 
known  ideas  and  bring  them  clearly  before  his  mind  before  he  attacks 
the  new  notions  and  attempts  to  get  results. 

d.  Learners,  who  without. aid  from  others  in  the  class  come  pre- 
pared from  day  to  day,  can  look  forward  with  much  pleasure  to  the 
study  of  higher  branches  of  mathematics.  Independent  work  and 
clearly  stated  analyses  go  along  together.  The  best  students  in  oral 
arithmetic  are  they  who  can  give  clearest  expression  to  the  solution 
of  the  problems  of  this  section. 

1.  The  ratio  of  two  squares  is  T9g.     If  the  larger  is  a 
25-in.  square,  what  is  the  other  ? 

2.  A  3-in.  circle  is  56\%  of  how  large  a  circle  ? 

134 


EXTENDING  PREVIOUS  IDEAS  135 

3.  What  are  the  dimensions  of  a  cube  that  is  337|  %  of 
a  25-in.  cube  ?     (3371%  =  g§ .) 

4.  What  is  the  diameter  of  a  sphere  that  is  ^Y  oi  a 
37^-in.  sphere  ? 

5.  How  many  blocks  8^  in.  x  5^  in.  x  4  in.,  can  be 
packed  into  a  box  25J  in.  x  16^  in.  x  8  in.  ? 

6.  What  may  be  the  dimensions  of  a  box  that  contains 
48  cu.  in.  ? 

7.  What  per  cent  of  the  amount  of  water  supplied  by  a 
4-in.  water  pipe  will  two  2-in.  pipes  supply  ? 

8.  Which  is  larger  and  how  much,  a  cone  9  in.  high 
whose  base  is  a  2-in.  circle,  or  a  pyramid  9  in.  high  whose 
base  is  a  2-in.  square  ? 

9.  How  many  \  -in.  water  pipes  have  the  same  capacity 
as  a  2-in.  main  ? 

10.  What  is  the  value  of  a  3-in.  iron  ball  if  a  2-in.  iron 
ball  is  worth  $.80? 

11.  If  a  block  of  wood  that  is  6^  in.  long,  5  in.  wide, 
and  4  in.  thick  weighs  3^  lb.,  how  much  does  a  block  of 
the  same  wood  that  is  18|  in.  long,  3  in.  wide,  and  2  in. 
thick  weigh  ? 

12.  If  a  timber  12  in.  x  12  in.  and  20  ft.  long  weighs 
600  lb.,  what  is  the  weight  of  one  of  the  same  kind 
24  in.  x  24  in.  and  30  ft.  long  ?  What  per  cent  of  the 
weight  of  the  former  is  the  latter  ? 

13.  Which  is  larger  and  how  much,  a  square  field  in- 
closed with  44  rd.  of  fence,  or  a  circular  field  requiring 
the  same  amount  of  fence  ? 


136  EXTENDING  PREVIOUS  IDEAS 

14.  How  many  gallons  of  water  (231  cu.  in.  to  the 
gallon)  will  a  3J-in.  pipe  carry  in  10  min.,  if  it  runs  at 
the  rate  of  5  ft.  per  second  ? 

How  many  cubic  inches  are  there  in  a  section  of  the  pipe  1  in. 
long?  8  in.  long?  How  many  gallons  in  a  section  2  ft.  long?  5  ft. 
long  ?  How  many  gallons  are  carried  per  second  ?  Per  minute  ?  In 
10  min.  ? 

15.  What  per  cent  of  a  25-in.  sphere  is  a  37^-in.  sphere  ? 

16.  There  are  two  iron  cylinders  of  the  same  weight, 
one  of  which  is  2  in.  in  diameter,  the  other  3  in.  What  is 
the  ratio  of  the  height  of  the  first  to  the  height  of  the 
second  ? 

17.  Two  square  iron  prisms  have  the  same  weight.  One 
is  \  the  height  of  the  other.  What  is  the  ratio  of  the  side 
of  the  base  of  one  to  the  side  of  the  base  of  the  other  ? 

18.  What  per  cent  of  a  trapezoid  whose  parallel  sides 
are  21  in.  and  9  in.,  and  altitude  8  in.,  is  the  smaller  of 
the  two  triangles  into  which  the  trapezoid  is  divided  by  a 
diagonal  ?  What  per  cent  of  it  is  the  larger  triangle  ? 
What  per  cent  of  the  smaller  triangle  is  the  larger  ? 

19.  What  is  the  diameter  of  a  sphere  which  is  T|g  of  a 
62J-in.  sphere  ? 

20.  What  is  the  side  of  a  cube  which  is  yy/s  of  a  7  2-in. 
cube? 

21.  What  is  the  diameter  of  a  cylinder  which  is  800% 
of  a  2-in.  cylinder  of  J  the  height  ? 

22.  Which  weighs  more,  a  2-in.  sphere  of  iron  or  a 
lf-in.  cube  of  iron  ? 

23.  How  many  4-in.  cylinders  6  in.  long  can  be  packed 
into  a  box  10  in.  x  10  in.  x  10  in.  ? 


EXTENDING  PREVIOUS  IDEAS  137 

24.  How  many  railroad  ties  8  in.  wide,  placed  22  in. 
apart  are  needed  for  5000  ft.  of  track  ? 

25.  A  hip  roof  has  two  equal  trapezoidal  portions  whose 
bases  are  40  ft.  and  20  ft.,  and  altitudes  20  ft.,  and  two 
equal  triangular  portions  whose  bases  are  30  ft.  and  alti- 
tudes 16  ft.  How  many  shingles  will  cover  the  roof, 
allowing  1000  shingles  to  each  100  sq.  ft.  ? 

26.  What  per  cent  of  a  cone  18 J  in.  high,  whose  base  is 
8j-  in.  in  diameter,  is  a  cone  121  in#  high,  whose  base  is 
25  in.  in  diameter  ? 

27.  How  many  revolutions  does  a  wagon  wheel  3J  ft.  in 
diameter  make  in  going  1  mile  ? 

How  many  revolutions  to  each  rod  ? 

28.  What  per  cent  of  the  area  of  a  circle  whose  circum- 
ference is  8  ft.  is  the  area  of  a  circle  whose  circumference 
is  6  ft.  ? 

29.  How  many  pounds  does  a  block  of  wood  2  ft.  long, 
18  in.  wide,  and  6  in.  thick  weigh,  if  a  3-in.  cube  of  the 
same  wood  weighs  \  lb.  ? 

30.  What  is  the  diameter  of  a  circle  whose  area  is  64% 
of  that  of  a  31^-in.  circle? 

31.  What  is  the  diameter  of  a  circle  whose  circumference 
is  60%  of  that  of  a  41f-in.  circle?  (See  problem  47, 
page  77.) 

32.  How  many  cubic  inches  are  there  in  a  piece  of  round 
iron  2  in.  in  diameter  and  7  ft.  long? 

33.  A  has  a  piece  of  land  42  rd.  square.  A  strip  1  rd. 
wide  is  taken  on  each  of  the  four  sides  for  street  purposes. 
How  many  square  rods  of  land  are  taken? 


138  EXTENDING  PREVIOUS  IDEAS 

34.  How  many  4-in.  cubes  equal  eight  10-in.  cubes? 

35.  What  is  the  value  of  a  4-ft.  cube  of  ice  weighing 
50  lb.  to  the  cubic  foot,  at  \9  per  pound? 

36.  What  is  the  greatest  number  of  blocks  9  in.  long  and 
2  in.  square  that  can  be  put  into  a  box  18  in.  x  12  in.  and 
11  in.  deep?     (An$.  66.)     How  must  they  be  arranged? 

37.  What  is  the  weight  of  a  wooden  cone  21  in.  high, 
having  a  2-in.  base,  if  a  cubic  inch  of  the  wood  weighs 
loz.? 

38.  How  many  cubic  inches  are  there  in  a  tin  pail  7  in. 
in  diameter  and  6  in.  deep?  How  many  gallons  will  a 
pail  hold  that  is  14  in.  in  diameter  and  16  in.  deep? 

39.  How  many  revolutions  does  a  3J-ft.  wheel  make  in 
running  around  a  circle  21  rd.  in  diameter? 

40.  How  many  times  must  you  dip  in  order  to  fill  a 
cylindrical  pail  10|  in.  in  diameter  and  9  in.  deep  with  a 
Conical  dipper  3  J  in.  across  at  the  top  and  3  in.  deep? 

How  high  is  the  cylinder  3£  in.  in  diameter,  that  is  equal  to  the 
3£-in.  cone? 

41.  What  is  the  weight  of  a  sphere  that  is  60%  of  a 
9-lb.  sphere  ?     Need  the  dimensions  be  known  ? 

42.  How  many  2|-in.  cubes  can  be  put  into  a  box  12J  in. 
each  way?     Into  one  30  in.  each  way? 

43.  How  many  cubic  yards  of  earth  are  removed  in 
digging  a  ditch  90  ft.  long,  1J  ft.  wide,  and  3  ft.  deep? 

44.  A's  land  measures  80  rd.  on  each  side,  B's  60  rd.  on 
each  side.  What  is  the  ratio  of  A's  land  to  B's?  What 
is  the  ratio  of  the  amount  of  fence  needed  to  inclose  A's 
to  the  amount  needed  to  inclose  B's? 


EXTENDING  PREVIOUS  IDEAS  139 

45.  Over  how  many  square  rods  of  land  can  a  horse  feed 
when  he  is  tied  with  a  rope  57f  ft.  long? 

How  many  rods  are  there  in  57£  ft.  ? 

46.  If  a  pipe  12J  in.  in  diameter  carries  2500  gal.  a 
minute,  how  many  gallons  a  minute  will  a  10-in.  pipe 
carry,  if  the  water  runs  at  the  same  speed? 

47.  In  boring  a  well  88  ft.  deep  with  a  2-ft.  auger,  how 
many  cubic  feet  of  earth  were  removed? 

48.  A  piece  of  sheet  iron  is  cut  so  that  two  sides, 
measuring  8  ft.  and  10  ft.  respectively,  are  parallel,  and 
the  distance  between  them  is  12  ft.  How  much  does  it 
weigh  at  the  rate  of  2  lb.  to  the  square  foot? 

49.  How  many  gallons  (7|-  gal.  to  1  cu.  ft.)  can  be  put 
into  a  tank  10  ft.  x  10  ft.  x  4  ft.  ? 

50.  What  is  the  diameter  of  the  base  of  a  cone  whose 
volume  is  33  cu.  in.,  and  whose  height  is  14  in.  ? 

51.  How  many  bushels  of  wheat  (1^  cu.  ft.  to  the  bushel) 
will  a  bin  8  ft.  x  10  ft.  x  4  ft.  hold? 

52.  How  many  times  can  a  cylindrical  pail  6  in.  in 
diameter  and  9  in.  deep  be  filled  from  a  pail  9  in.  in 
diameter  and  12  in.  deep? 

53.  What  is  the  ratio  of  a  strip  2  rd.  wide  around  an 
80-rd.  square,  to  a  strip  2  rd.  wide  around  a  40-rd.  square? 

54.  How  many  feet  of  lumber  are  necessary  to  cover 
with  inch  boards  a  wall  30  ft.  long  and  20  ft.  high? 

55.  At  $20  per  M.,  what  is  the  value  of  five  8  x  8's, 
18  ft.  long,  and  ten  2  x  6's  12  ft.  long? 

56.  Of  two  cylinders  of  equal  diameter,  one  is  |  as  long 
as  the  other.     What  is  the  ratio  of  their  volumes  ? 


140  EXTENDING  PREVIOUS  IDEAS 

57.  Of  two  cylinders  of  equal  heights,  one  has  a  diameter 
equal  to  J  of  the  diameter  of  the  other.  What  is  the  ratio 
of  their  volumes? 

58.  Of  two  cylinders  of  the  same  material  and  of  the 
same  height,  one  measures  25  in.  around,  and  the  other 
31  \  in.  If  the  first  weighs  128  lb.,  how  much  does  the 
other  weigh? 

59.  Which  is  larger,  a  square  field  88  rd.  around,  or  a 
circular  field  88  rd.  around? 

60.  What  may  be  the  dimensions  of  a  box  in  order  that 
it  may  exactly  hold  320  blocks  2  in.  by  5  in.  by  8  in.  ? 

61.  How  many  3|-in.  spheres  weigh  as  much  as  a  14-in. 
sphere  of  the  same  material? 

62.  At  1 2  per  M.  for  shingles  and  $  1  per  M.  for  laying, 
what  is  the  cost  of  shingling  a  roof  consisting  of  four  equal 
triangular  portions,  the  base  of  each  being  20  ft.  and  alti- 
tude 15  ft.? 

63.  How  many  cubic  inches  of  wood  are  removed  in 
boring  through  a  timber  10J  in.  thick  with  a  2-in.  auger? 

64.  What  per  cent  of  A's  land  is  B's,  if  A's  is  160  rd. 
long  and  80  rd.  wide,  and  B's  is  80  rd.  long  and  60  rd.  wide? 

65.  At  what  price  must  a  merchant  mark  an  article 
costing  $1.50  so  as  to  gain  20%  after  making  a  discount 
of  10%  to  his  customer? 

66.  At  what  price  must  I  buy  United  States  4%  bonds 
to  realize  3%  on  my  investment? 

67.  I  sold  two  horses  for  $  96  each.  On  one  I  gained 
20%  and  on  the  other  I  lost  20%.  What  was  my  gain  or 
loss?     What  was  the  per  cent  of  gain  or  loss? 


EXTENDING  PREVIOUS  IDEAS  141 

68.  A  man's  house  cost  18000.  He  insured  |  of  its 
value  at  |%.      The  house  burned.     What  was  his  loss? 

69.  A  man  paid  $  15  premium  on  a  barn.  The  rate  was 
f%.  When  the  barn  burned,  how  much  was  his  loss  if 
the  barn  was  insured  for  ^  of  its  cost? 

70.  A,  B,  and  C  earn  $270.  A  earns  twice  as  much  as 
B,  and  C  earns  as  much  as  the  other  two.  What  per  cent 
of  the  whole  does  each  one  earn  ? 

71.  80%  of  the  cost  of  a  house  equals  f  of  the  selling 
price.     What  is  the  gain  per  cent? 


Arithmetic   Blanks 


Arithmetic  blanks  with  graded  examples  are  a  most 
convenient,  economical,  and  useful  aid  in  class  room'work. 
They  assist  the  teacher  by  furnishing  a  large  number  of 
carefully  classified  and  graded  examples  which  may  be 
used  for  regular  class  drills  and  for  examination  tests. 
The  examples,  being  without  answers,  furnish  a  .uniform 
standard  of  comparison  and  a  complete  test  of  the  pupil's 
progress.  The  best  and  cheapest  arithmetic  blanks  are 
the  following: 

NATIONAL  NUMBER  TABLETS 

Twelve  numbers Per  dozen  90  cents 

This  series  comprises  twelve  tablets  or  numbers  and  supplies  suf- 
ficient work  to  cover  the  whole  course  of  written  arithmetic.  The  tablets 
and  lessons  are  carefully  graded  and  so  arranged  that  two  tablets  furnish 
enough  supplementary  work  for  a  school  year. 

RAY'S  TEST  EXAMPLE  TABLETS 

Eight  numbers    .         .         .         .         .         .         .     Per  dozen  $1.00 

These  tablets  furnish  in  convenient  form  well  selected  and  carefully 

graded  test  examples,  each  sheet  having  printed  at  the  head  from  five  to 

ten  problems.     The  eight  numbers  cover  a  full  course  of  arithmetical 

operations. 

SILVER'S  PRIMARY  EXERCISES  IN  ARITHMETIC 

Numbers  i  and  2 Each  10  cents 

Numbers  3  and  4 Each  15  cents 

A  series  of  graded  exercises  in  the  fundamental  rules  of  arithmetic 
for  beginners;  one  page  for  each  school  day,  printed  in  large,  bold  type, 
giving  the  pupil  a  large  amount  of  practice.  The  answers  to  the  examples 
are  to  be  recorded  by  the  pupil  on  the  printed  page.  These  blanks  will 
be  found  a  very  useful  supplement  to  any  text-book  in  arithmetic. 


Specimen  copies  of  any  of  the  above  Arithmetic  Blanks  will  be  sent, 
prepaid,  to  any  address  on  receipt  of  the  price. 

American   Book  Company 

New  York  ♦  Cincinnati  ♦  Chicago 

(48) 


Spelling  and  Word   Study 


RICE'S  RATIONAL  SPELLING  BOOK.     By  Dr.  J.  M.  Rice. 

Part  I.  For  use  in  the  first  three  grades.  Boards  .  .15  cents 
The  Same.     Cloth    .         .       . 17  cents 

Part  II.  For  use  in  all  grades  above  the  third.  Boards.  20  cents 
The  Same.     Cloth 22  cents 

This  is  a  spelling  book  pure  and  simple,  designed  for  use  in  all  the 
grades  or  years  of  school  work.  It  has  been  arranged  on  a  definite 
psychological  plan,  based  upon  an  examination  of  the  schools  of  nearly 
all  the  large  cities  of  the  country  and  upon  a  careful  study  of  the  actual 
spelling  of  many  thousand  pupils. 

PATTERSON'S  AMERICAN  WORD  BOOK 

Graded  Lessons  in  Spelling,  Defining,  Punctuation,  and  Dictation. 

By  Calvin  Patterson,  M.A 25  cents 

This  new  spelling  book  embodies  a  carefully  developed  and  pro- 
gressive plan  for  teaching  the  forms  and  values  of  English  words  in 
common  use.  It  begins  with  words  illustrating  the  primary  sounds,  the 
words  being  printed  both  in  Roman  letters  and  in  vertical  script.  Then 
follow  graded  lessons  on  different  classes  and  uses  of  words. 

HARRINGTON'S  SPELLING  BOOK.     Complete  .        .        .     20  cents 
Parti. — separate.     For  Primary  Grades  .         .         .         .     15  cents 

Part  II. — separate.     For  Higher  Grades 15  cents 

A  complete  course  in  spelling  for  graded  and  common  schools. 

McGUFFEY'S  REVISED  ECLECTIC  SPELLING  BOOK      .      17  cents 
Conforming    in    orthography,    pronunciation,    syllabication,    and 
diacritical  marks  to  the  Websterian  standard. 

NATURAL  SPELLER  AND  WORD  BOOK  ....      20  cents 

SWINTON'S  WORD  BOOK  OF  ENGLISH  SPELLING       .       18  cents 


Copies  of  any  of  the  above  books  will  be  sent,  prepaid,  to  any  address  on 
receipt  of  the  price  by  the  Publishers  : 

American    Book    Company 

New  York  ♦  Cincinnati  ♦  Chicago 

(32) 


Maxwell's    English    Course 

By   WILLIAM    H.    MAXWELL,    M.A.,    Ph.D. 
Superintendent  of  Schools,  City  of  New  York. 

FIRST  BOOK  IN  ENGLISH 

For  use  in  Elementary  Grades         .         .         .         .         .40  cents 

INTRODUCTORY  LESSONS  IN  ENGLISH  GRAMMAR 

For  use  in  Intermediate  and  Grammar  Grades    .     .         .40  cents 
(These  two  books  constitute  a  complete  graded  course  in  English 
for  Elementary  and  Grammar  Grades.) 


ADVANCED  LESSONS  IN  ENGLISH  GRAMMAR 

For  Higher  Grammar  Classes  and  High  Schools     .         .     60  cents 


The  "First  Book  in  English  "  combines  lessons,  prac- 
tice, and  instruction  in  the  elementary  principles  in  the 
English  language,  in  such  a  rational  and  practical  way  as 
to  make  a  text-book  for  beginners  in  language  study,  which 
avoids  the  platitudes  of  modern  ' '  language  lessons  "  on  one 
hand,  and  the  difficulties  of  "technical "  grammar  on  the 
other. 

The  "Introductory  Lessons"  presents  as  much  of  the 
science  of  grammar  with  its  applications  as  children  can 
understand  and  appreciate,  before  taking  up  an  advanced 
course  in  English.  The  book  contains  in  a  compact  form 
a  well-graded  and  perspicuous  treatment  of  all  the  subjects 
usually  taught  in  English  Grammar.  It  omits  no  essen- 
tial principle  or  definition  or  example,  and  is  sufficiently 
complete  to  meet  all  the  requirements  of  the  usual  course 
of  study  of  Intermediate  or  Grammar  Schools. 

The  "Advanced  Lessons  in  English  Grammar"  em- 
braces all  the  theory  and  all  the  practice  that  are  necessary 
during  the  last  two  years  of  a  grammar-school  or  through- 
out a  high  school  course.  It  is  intended  to  serve  two 
purposes :  First,  that  of  a  text-book,  supplying  the 
principles  and  rules  of  the  science  as  well  as  their  applica- 
tion in  copious  exercises;  Second,  a  book  of  reference,  to 
be  used  whenever  difficulties  are  presented  either  in  the 
student's  own  compositions  or  in  literature  that  is  sub- 
jected to  critical  study. 

Copies  sent  to  any  address,  prepaid \  on  receipt  of  price. 

American   Book  Company 
New  York  ♦  Cincinnati  •  Chicago 

(78) 


YB  35876 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


■ 


J 


